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Properties

Label 63.4.e.c.37.3
Level $63$
Weight $4$
Character 63.37
Analytic conductor $3.717$
Analytic rank $0$
Dimension $6$
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] = [63,4,Mod(37,63)] 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("63.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.9924270768.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 25x^{4} + 12x^{3} + 582x^{2} - 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.3
Root \(2.65415 - 4.59712i\) of defining polynomial
Character \(\chi\) \(=\) 63.37
Dual form 63.4.e.c.46.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.65415 - 4.59712i) q^{2} +(-10.0890 - 17.4746i) q^{4} +(2.78070 - 4.81631i) q^{5} +(9.67799 + 15.7904i) q^{7} -64.6443 q^{8} +(-14.7608 - 25.5664i) q^{10} +(-6.95869 - 12.0528i) q^{11} +38.6718 q^{13} +(98.2771 - 2.58082i) q^{14} +(-90.8636 + 157.380i) q^{16} +(21.7394 + 37.6537i) q^{17} +(54.5139 - 94.4208i) q^{19} -112.218 q^{20} -73.8775 q^{22} +(-37.4389 + 64.8461i) q^{23} +(47.0354 + 81.4677i) q^{25} +(102.641 - 177.779i) q^{26} +(178.290 - 328.429i) q^{28} +72.3589 q^{29} +(-32.0215 - 55.4629i) q^{31} +(223.754 + 387.553i) q^{32} +230.798 q^{34} +(102.963 - 2.70387i) q^{35} +(-94.3636 + 163.443i) q^{37} +(-289.376 - 501.213i) q^{38} +(-179.756 + 311.347i) q^{40} +24.7923 q^{41} -243.881 q^{43} +(-140.412 + 243.201i) q^{44} +(198.737 + 344.222i) q^{46} +(-310.274 + 537.411i) q^{47} +(-155.673 + 305.638i) q^{49} +499.356 q^{50} +(-390.159 - 675.776i) q^{52} +(-143.919 - 249.276i) q^{53} -77.4001 q^{55} +(-625.627 - 1020.76i) q^{56} +(192.051 - 332.642i) q^{58} +(262.526 + 454.708i) q^{59} +(191.718 - 332.065i) q^{61} -339.960 q^{62} +921.681 q^{64} +(107.535 - 186.255i) q^{65} +(-99.0583 - 171.574i) q^{67} +(438.657 - 759.776i) q^{68} +(260.849 - 480.510i) q^{70} -785.432 q^{71} +(165.570 + 286.776i) q^{73} +(500.910 + 867.602i) q^{74} -2199.96 q^{76} +(122.972 - 226.527i) q^{77} +(-218.823 + 379.013i) q^{79} +(505.329 + 875.255i) q^{80} +(65.8024 - 113.973i) q^{82} -241.241 q^{83} +241.803 q^{85} +(-647.297 + 1121.15i) q^{86} +(449.840 + 779.145i) q^{88} +(792.772 - 1373.12i) q^{89} +(374.265 + 610.643i) q^{91} +1510.88 q^{92} +(1647.03 + 2852.73i) q^{94} +(-303.173 - 525.112i) q^{95} +79.2754 q^{97} +(991.877 + 1526.86i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 25 q^{4} + 11 q^{5} - 13 q^{7} - 78 q^{8} + 55 q^{10} + 35 q^{11} + 124 q^{13} + 326 q^{14} - 241 q^{16} + 48 q^{17} + 202 q^{19} - 878 q^{20} - 14 q^{22} + 216 q^{23} - 130 q^{25} + 274 q^{26}+ \cdots + 4267 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.65415 − 4.59712i 0.938383 − 1.62533i 0.169895 − 0.985462i \(-0.445657\pi\)
0.768488 − 0.639864i \(-0.221009\pi\)
\(3\) 0 0
\(4\) −10.0890 − 17.4746i −1.26112 − 2.18433i
\(5\) 2.78070 − 4.81631i 0.248713 − 0.430784i −0.714456 − 0.699681i \(-0.753326\pi\)
0.963169 + 0.268897i \(0.0866590\pi\)
\(6\) 0 0
\(7\) 9.67799 + 15.7904i 0.522562 + 0.852601i
\(8\) −64.6443 −2.85690
\(9\) 0 0
\(10\) −14.7608 − 25.5664i −0.466777 − 0.808481i
\(11\) −6.95869 − 12.0528i −0.190738 − 0.330369i 0.754757 − 0.656005i \(-0.227755\pi\)
−0.945495 + 0.325636i \(0.894422\pi\)
\(12\) 0 0
\(13\) 38.6718 0.825048 0.412524 − 0.910947i \(-0.364647\pi\)
0.412524 + 0.910947i \(0.364647\pi\)
\(14\) 98.2771 − 2.58082i 1.87612 − 0.0492680i
\(15\) 0 0
\(16\) −90.8636 + 157.380i −1.41974 + 2.45907i
\(17\) 21.7394 + 37.6537i 0.310152 + 0.537198i 0.978395 − 0.206744i \(-0.0662869\pi\)
−0.668243 + 0.743943i \(0.732954\pi\)
\(18\) 0 0
\(19\) 54.5139 − 94.4208i 0.658228 − 1.14009i −0.322845 − 0.946452i \(-0.604639\pi\)
0.981074 − 0.193633i \(-0.0620273\pi\)
\(20\) −112.218 −1.25463
\(21\) 0 0
\(22\) −73.8775 −0.715943
\(23\) −37.4389 + 64.8461i −0.339415 + 0.587885i −0.984323 − 0.176376i \(-0.943563\pi\)
0.644907 + 0.764261i \(0.276896\pi\)
\(24\) 0 0
\(25\) 47.0354 + 81.4677i 0.376283 + 0.651742i
\(26\) 102.641 − 177.779i 0.774211 − 1.34097i
\(27\) 0 0
\(28\) 178.290 − 328.429i 1.20335 − 2.21668i
\(29\) 72.3589 0.463335 0.231667 − 0.972795i \(-0.425582\pi\)
0.231667 + 0.972795i \(0.425582\pi\)
\(30\) 0 0
\(31\) −32.0215 − 55.4629i −0.185524 − 0.321337i 0.758229 − 0.651988i \(-0.226065\pi\)
−0.943753 + 0.330652i \(0.892732\pi\)
\(32\) 223.754 + 387.553i 1.23608 + 2.14095i
\(33\) 0 0
\(34\) 230.798 1.16416
\(35\) 102.963 − 2.70387i 0.497255 − 0.0130582i
\(36\) 0 0
\(37\) −94.3636 + 163.443i −0.419278 + 0.726211i −0.995867 − 0.0908235i \(-0.971050\pi\)
0.576589 + 0.817034i \(0.304383\pi\)
\(38\) −289.376 − 501.213i −1.23534 − 2.13967i
\(39\) 0 0
\(40\) −179.756 + 311.347i −0.710550 + 1.23071i
\(41\) 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 \(0.642354\pi\)
\(44\) −140.412 + 243.201i −0.481090 + 0.833272i
\(45\) 0 0
\(46\) 198.737 + 344.222i 0.637003 + 1.10332i
\(47\) −310.274 + 537.411i −0.962940 + 1.66786i −0.247888 + 0.968789i \(0.579736\pi\)
−0.715052 + 0.699071i \(0.753597\pi\)
\(48\) 0 0
\(49\) −155.673 + 305.638i −0.453857 + 0.891074i
\(50\) 499.356 1.41239
\(51\) 0 0
\(52\) −390.159 − 675.776i −1.04049 − 1.80218i
\(53\) −143.919 − 249.276i −0.372997 − 0.646050i 0.617028 − 0.786941i \(-0.288337\pi\)
−0.990025 + 0.140891i \(0.955003\pi\)
\(54\) 0 0
\(55\) −77.4001 −0.189757
\(56\) −625.627 − 1020.76i −1.49291 − 2.43580i
\(57\) 0 0
\(58\) 192.051 − 332.642i 0.434785 − 0.753070i
\(59\) 262.526 + 454.708i 0.579287 + 1.00335i 0.995561 + 0.0941152i \(0.0300022\pi\)
−0.416275 + 0.909239i \(0.636664\pi\)
\(60\) 0 0
\(61\) 191.718 − 332.065i 0.402409 − 0.696993i −0.591607 − 0.806226i \(-0.701506\pi\)
0.994016 + 0.109234i \(0.0348397\pi\)
\(62\) −339.960 −0.696369
\(63\) 0 0
\(64\) 921.681 1.80016
\(65\) 107.535 − 186.255i 0.205200 − 0.355417i
\(66\) 0 0
\(67\) −99.0583 − 171.574i −0.180625 − 0.312852i 0.761468 − 0.648202i \(-0.224479\pi\)
−0.942094 + 0.335350i \(0.891145\pi\)
\(68\) 438.657 − 759.776i 0.782279 − 1.35495i
\(69\) 0 0
\(70\) 260.849 − 480.510i 0.445392 − 0.820456i
\(71\) −785.432 −1.31287 −0.656434 − 0.754384i \(-0.727936\pi\)
−0.656434 + 0.754384i \(0.727936\pi\)
\(72\) 0 0
\(73\) 165.570 + 286.776i 0.265459 + 0.459789i 0.967684 − 0.252166i \(-0.0811430\pi\)
−0.702224 + 0.711956i \(0.747810\pi\)
\(74\) 500.910 + 867.602i 0.786887 + 1.36293i
\(75\) 0 0
\(76\) −2199.96 −3.32043
\(77\) 122.972 − 226.527i 0.182000 − 0.335262i
\(78\) 0 0
\(79\) −218.823 + 379.013i −0.311640 + 0.539776i −0.978718 − 0.205212i \(-0.934212\pi\)
0.667078 + 0.744988i \(0.267545\pi\)
\(80\) 505.329 + 875.255i 0.706219 + 1.22321i
\(81\) 0 0
\(82\) 65.8024 − 113.973i 0.0886178 − 0.153491i
\(83\) −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
\(86\) −647.297 + 1121.15i −0.811626 + 1.40578i
\(87\) 0 0
\(88\) 449.840 + 779.145i 0.544921 + 0.943831i
\(89\) 792.772 − 1373.12i 0.944198 − 1.63540i 0.186849 − 0.982389i \(-0.440172\pi\)
0.757349 − 0.653010i \(-0.226494\pi\)
\(90\) 0 0
\(91\) 374.265 + 610.643i 0.431139 + 0.703437i
\(92\) 1510.88 1.71218
\(93\) 0 0
\(94\) 1647.03 + 2852.73i 1.80721 + 3.13018i
\(95\) −303.173 − 525.112i −0.327420 − 0.567109i
\(96\) 0 0
\(97\) 79.2754 0.0829814 0.0414907 − 0.999139i \(-0.486789\pi\)
0.0414907 + 0.999139i \(0.486789\pi\)
\(98\) 991.877 + 1526.86i 1.02239 + 1.57384i
\(99\) 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 63.4.e.c.37.3 6
3.2 odd 2 21.4.e.b.16.1 yes 6
7.2 even 3 441.4.a.s.1.1 3
7.3 odd 6 441.4.e.w.361.3 6
7.4 even 3 inner 63.4.e.c.46.3 6
7.5 odd 6 441.4.a.t.1.1 3
7.6 odd 2 441.4.e.w.226.3 6
12.11 even 2 336.4.q.k.289.2 6
21.2 odd 6 147.4.a.l.1.3 3
21.5 even 6 147.4.a.m.1.3 3
21.11 odd 6 21.4.e.b.4.1 ✓ 6
21.17 even 6 147.4.e.n.67.1 6
21.20 even 2 147.4.e.n.79.1 6
84.11 even 6 336.4.q.k.193.2 6
84.23 even 6 2352.4.a.ci.1.2 3
84.47 odd 6 2352.4.a.cg.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.b.4.1 ✓ 6 21.11 odd 6
21.4.e.b.16.1 yes 6 3.2 odd 2
63.4.e.c.37.3 6 1.1 even 1 trivial
63.4.e.c.46.3 6 7.4 even 3 inner
147.4.a.l.1.3 3 21.2 odd 6
147.4.a.m.1.3 3 21.5 even 6
147.4.e.n.67.1 6 21.17 even 6
147.4.e.n.79.1 6 21.20 even 2
336.4.q.k.193.2 6 84.11 even 6
336.4.q.k.289.2 6 12.11 even 2
441.4.a.s.1.1 3 7.2 even 3
441.4.a.t.1.1 3 7.5 odd 6
441.4.e.w.226.3 6 7.6 odd 2
441.4.e.w.361.3 6 7.3 odd 6
2352.4.a.cg.1.2 3 84.47 odd 6
2352.4.a.ci.1.2 3 84.23 even 6