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Properties

Label 147.4.a.m.1.3
Level $147$
Weight $4$
Character 147.1
Self dual yes
Analytic conductor $8.673$
Analytic rank $0$
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] = [147,4,Mod(1,147)] 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("147.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,1,9,25,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.57516.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 24x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(5.30829\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.30829 q^{2} +3.00000 q^{3} +20.1780 q^{4} -5.56140 q^{5} +15.9249 q^{6} +64.6443 q^{8} +9.00000 q^{9} -29.5215 q^{10} -13.9174 q^{11} +60.5340 q^{12} -38.6718 q^{13} -16.6842 q^{15} +181.727 q^{16} -43.4788 q^{17} +47.7746 q^{18} +109.028 q^{19} -112.218 q^{20} -73.8775 q^{22} -74.8778 q^{23} +193.933 q^{24} -94.0708 q^{25} -205.281 q^{26} +27.0000 q^{27} -72.3589 q^{29} -88.5646 q^{30} -64.0431 q^{31} +447.507 q^{32} -41.7521 q^{33} -230.798 q^{34} +181.602 q^{36} +188.727 q^{37} +578.751 q^{38} -116.015 q^{39} -359.513 q^{40} +24.7923 q^{41} -243.881 q^{43} -280.825 q^{44} -50.0526 q^{45} -397.474 q^{46} +620.549 q^{47} +545.182 q^{48} -499.356 q^{50} -130.436 q^{51} -780.319 q^{52} -287.839 q^{53} +143.324 q^{54} +77.4001 q^{55} +327.083 q^{57} -384.102 q^{58} -525.051 q^{59} -336.654 q^{60} +383.436 q^{61} -339.960 q^{62} +921.681 q^{64} +215.069 q^{65} -221.633 q^{66} +198.117 q^{67} -877.314 q^{68} -224.634 q^{69} +785.432 q^{71} +581.799 q^{72} +331.141 q^{73} +1001.82 q^{74} -282.213 q^{75} +2199.96 q^{76} -615.844 q^{78} +437.647 q^{79} -1010.66 q^{80} +81.0000 q^{81} +131.605 q^{82} -241.241 q^{83} +241.803 q^{85} -1294.59 q^{86} -217.077 q^{87} -899.680 q^{88} -1585.54 q^{89} -265.694 q^{90} -1510.88 q^{92} -192.129 q^{93} +3294.05 q^{94} -606.347 q^{95} +1342.52 q^{96} -79.2754 q^{97} -125.256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 9 q^{3} + 25 q^{4} - 11 q^{5} + 3 q^{6} + 39 q^{8} + 27 q^{9} + 55 q^{10} + 35 q^{11} + 75 q^{12} - 62 q^{13} - 33 q^{15} + 241 q^{16} - 48 q^{17} + 9 q^{18} + 202 q^{19} - 439 q^{20} - 7 q^{22}+ \cdots + 315 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.30829 1.87677 0.938383 − 0.345598i \(-0.112324\pi\)
0.938383 + 0.345598i \(0.112324\pi\)
\(3\) 3.00000 0.577350
\(4\) 20.1780 2.52225
\(5\) −5.56140 −0.497427 −0.248713 − 0.968577i \(-0.580008\pi\)
−0.248713 + 0.968577i \(0.580008\pi\)
\(6\) 15.9249 1.08355
\(7\) 0 0
\(8\) 64.6443 2.85690
\(9\) 9.00000 0.333333
\(10\) −29.5215 −0.933553
\(11\) −13.9174 −0.381477 −0.190738 − 0.981641i \(-0.561088\pi\)
−0.190738 + 0.981641i \(0.561088\pi\)
\(12\) 60.5340 1.45622
\(13\) −38.6718 −0.825048 −0.412524 − 0.910947i \(-0.635353\pi\)
−0.412524 + 0.910947i \(0.635353\pi\)
\(14\) 0 0
\(15\) −16.6842 −0.287189
\(16\) 181.727 2.83949
\(17\) −43.4788 −0.620303 −0.310152 − 0.950687i \(-0.600380\pi\)
−0.310152 + 0.950687i \(0.600380\pi\)
\(18\) 47.7746 0.625588
\(19\) 109.028 1.31646 0.658228 − 0.752818i \(-0.271306\pi\)
0.658228 + 0.752818i \(0.271306\pi\)
\(20\) −112.218 −1.25463
\(21\) 0 0
\(22\) −73.8775 −0.715943
\(23\) −74.8778 −0.678831 −0.339415 − 0.940637i \(-0.610229\pi\)
−0.339415 + 0.940637i \(0.610229\pi\)
\(24\) 193.933 1.64943
\(25\) −94.0708 −0.752567
\(26\) −205.281 −1.54842
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −72.3589 −0.463335 −0.231667 − 0.972795i \(-0.574418\pi\)
−0.231667 + 0.972795i \(0.574418\pi\)
\(30\) −88.5646 −0.538987
\(31\) −64.0431 −0.371048 −0.185524 − 0.982640i \(-0.559398\pi\)
−0.185524 + 0.982640i \(0.559398\pi\)
\(32\) 447.507 2.47215
\(33\) −41.7521 −0.220246
\(34\) −230.798 −1.16416
\(35\) 0 0
\(36\) 181.602 0.840749
\(37\) 188.727 0.838556 0.419278 − 0.907858i \(-0.362283\pi\)
0.419278 + 0.907858i \(0.362283\pi\)
\(38\) 578.751 2.47068
\(39\) −116.015 −0.476342
\(40\) −359.513 −1.42110
\(41\) 24.7923 0.0944367 0.0472184 − 0.998885i \(-0.484964\pi\)
0.0472184 + 0.998885i \(0.484964\pi\)
\(42\) 0 0
\(43\) −243.881 −0.864920 −0.432460 − 0.901653i \(-0.642354\pi\)
−0.432460 + 0.901653i \(0.642354\pi\)
\(44\) −280.825 −0.962180
\(45\) −50.0526 −0.165809
\(46\) −397.474 −1.27401
\(47\) 620.549 1.92588 0.962940 − 0.269717i \(-0.0869303\pi\)
0.962940 + 0.269717i \(0.0869303\pi\)
\(48\) 545.182 1.63938
\(49\) 0 0
\(50\) −499.356 −1.41239
\(51\) −130.436 −0.358132
\(52\) −780.319 −2.08098
\(53\) −287.839 −0.745995 −0.372997 − 0.927832i \(-0.621670\pi\)
−0.372997 + 0.927832i \(0.621670\pi\)
\(54\) 143.324 0.361184
\(55\) 77.4001 0.189757
\(56\) 0 0
\(57\) 327.083 0.760057
\(58\) −384.102 −0.869571
\(59\) −525.051 −1.15857 −0.579287 − 0.815124i \(-0.696669\pi\)
−0.579287 + 0.815124i \(0.696669\pi\)
\(60\) −336.654 −0.724363
\(61\) 383.436 0.804818 0.402409 − 0.915460i \(-0.368173\pi\)
0.402409 + 0.915460i \(0.368173\pi\)
\(62\) −339.960 −0.696369
\(63\) 0 0
\(64\) 921.681 1.80016
\(65\) 215.069 0.410401
\(66\) −221.633 −0.413350
\(67\) 198.117 0.361251 0.180625 − 0.983552i \(-0.442188\pi\)
0.180625 + 0.983552i \(0.442188\pi\)
\(68\) −877.314 −1.56456
\(69\) −224.634 −0.391923
\(70\) 0 0
\(71\) 785.432 1.31287 0.656434 − 0.754384i \(-0.272064\pi\)
0.656434 + 0.754384i \(0.272064\pi\)
\(72\) 581.799 0.952301
\(73\) 331.141 0.530919 0.265459 − 0.964122i \(-0.414476\pi\)
0.265459 + 0.964122i \(0.414476\pi\)
\(74\) 1001.82 1.57377
\(75\) −282.213 −0.434495
\(76\) 2199.96 3.32043
\(77\) 0 0
\(78\) −615.844 −0.893981
\(79\) 437.647 0.623280 0.311640 − 0.950200i \(-0.399122\pi\)
0.311640 + 0.950200i \(0.399122\pi\)
\(80\) −1010.66 −1.41244
\(81\) 81.0000 0.111111
\(82\) 131.605 0.177236
\(83\) −241.241 −0.319032 −0.159516 − 0.987195i \(-0.550993\pi\)
−0.159516 + 0.987195i \(0.550993\pi\)
\(84\) 0 0
\(85\) 241.803 0.308555
\(86\) −1294.59 −1.62325
\(87\) −217.077 −0.267506
\(88\) −899.680 −1.08984
\(89\) −1585.54 −1.88840 −0.944198 − 0.329378i \(-0.893161\pi\)
−0.944198 + 0.329378i \(0.893161\pi\)
\(90\) −265.694 −0.311184
\(91\) 0 0
\(92\) −1510.88 −1.71218
\(93\) −192.129 −0.214224
\(94\) 3294.05 3.61442
\(95\) −606.347 −0.654841
\(96\) 1342.52 1.42730
\(97\) −79.2754 −0.0829814 −0.0414907 − 0.999139i \(-0.513211\pi\)
−0.0414907 + 0.999139i \(0.513211\pi\)
\(98\) 0 0
\(99\) −125.256 −0.127159
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 147.4.a.m.1.3 3
3.2 odd 2 441.4.a.t.1.1 3
4.3 odd 2 2352.4.a.cg.1.2 3
7.2 even 3 147.4.e.n.67.1 6
7.3 odd 6 21.4.e.b.16.1 yes 6
7.4 even 3 147.4.e.n.79.1 6
7.5 odd 6 21.4.e.b.4.1 ✓ 6
7.6 odd 2 147.4.a.l.1.3 3
21.2 odd 6 441.4.e.w.361.3 6
21.5 even 6 63.4.e.c.46.3 6
21.11 odd 6 441.4.e.w.226.3 6
21.17 even 6 63.4.e.c.37.3 6
21.20 even 2 441.4.a.s.1.1 3
28.3 even 6 336.4.q.k.289.2 6
28.19 even 6 336.4.q.k.193.2 6
28.27 even 2 2352.4.a.ci.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.b.4.1 ✓ 6 7.5 odd 6
21.4.e.b.16.1 yes 6 7.3 odd 6
63.4.e.c.37.3 6 21.17 even 6
63.4.e.c.46.3 6 21.5 even 6
147.4.a.l.1.3 3 7.6 odd 2
147.4.a.m.1.3 3 1.1 even 1 trivial
147.4.e.n.67.1 6 7.2 even 3
147.4.e.n.79.1 6 7.4 even 3
336.4.q.k.193.2 6 28.19 even 6
336.4.q.k.289.2 6 28.3 even 6
441.4.a.s.1.1 3 21.20 even 2
441.4.a.t.1.1 3 3.2 odd 2
441.4.e.w.226.3 6 21.11 odd 6
441.4.e.w.361.3 6 21.2 odd 6
2352.4.a.cg.1.2 3 4.3 odd 2
2352.4.a.ci.1.2 3 28.27 even 2