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Properties

Label 63.10.e.b.37.1
Level $63$
Weight $10$
Character 63.37
Analytic conductor $32.447$
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] = [63,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,18] 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: \(32.4472576783\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
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: \( x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + \cdots + 761760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.1
Root \(8.80924 - 15.2580i\) of defining polynomial
Character \(\chi\) \(=\) 63.37
Dual form 63.10.e.b.46.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-15.6185 + 27.0520i) q^{2} +(-231.874 - 401.617i) q^{4} +(239.755 - 415.269i) q^{5} +(1428.40 - 6189.77i) q^{7} -1507.26 q^{8} +(7489.23 + 12971.7i) q^{10} +(-33637.5 - 58261.9i) q^{11} +39620.0 q^{13} +(145136. + 135316. i) q^{14} +(142260. - 246402. i) q^{16} +(106607. + 184649. i) q^{17} +(-566407. + 981046. i) q^{19} -222372. q^{20} +2.10147e6 q^{22} +(-365109. + 632388. i) q^{23} +(861597. + 1.49233e6i) q^{25} +(-618804. + 1.07180e6i) q^{26} +(-2.81713e6 + 861574. i) q^{28} -383457. q^{29} +(2.73393e6 + 4.73530e6i) q^{31} +(4.05793e6 + 7.02853e6i) q^{32} -6.66015e6 q^{34} +(-2.22795e6 - 2.07720e6i) q^{35} +(3.83476e6 - 6.64200e6i) q^{37} +(-1.76928e7 - 3.06449e7i) q^{38} +(-361374. + 625919. i) q^{40} -1.08865e6 q^{41} -1.92137e7 q^{43} +(-1.55993e7 + 2.70188e7i) q^{44} +(-1.14049e7 - 1.97539e7i) q^{46} +(-2.11223e7 + 3.65850e7i) q^{47} +(-3.62729e7 - 1.76830e7i) q^{49} -5.38273e7 q^{50} +(-9.18684e6 - 1.59121e7i) q^{52} +(-2.68028e7 - 4.64237e7i) q^{53} -3.22591e7 q^{55} +(-2.15298e6 + 9.32961e6i) q^{56} +(5.98901e6 - 1.03733e7i) q^{58} +(9.06495e6 + 1.57010e7i) q^{59} +(-1.48303e7 + 2.56868e7i) q^{61} -1.70799e8 q^{62} -1.07840e8 q^{64} +(9.49911e6 - 1.64529e7i) q^{65} +(8.34811e7 + 1.44594e8i) q^{67} +(4.94387e7 - 8.56303e7i) q^{68} +(9.09897e7 - 2.78278e7i) q^{70} +3.15617e8 q^{71} +(1.51081e7 + 2.61679e7i) q^{73} +(1.19786e8 + 2.07476e8i) q^{74} +5.25340e8 q^{76} +(-4.08676e8 + 1.24987e8i) q^{77} +(-2.62085e8 + 4.53944e8i) q^{79} +(-6.82155e7 - 1.18153e8i) q^{80} +(1.70030e7 - 2.94500e7i) q^{82} +6.01000e8 q^{83} +1.02238e8 q^{85} +(3.00089e8 - 5.19769e8i) q^{86} +(5.07006e7 + 8.78160e7i) q^{88} +(-4.80628e8 + 8.32473e8i) q^{89} +(5.65933e7 - 2.45239e8i) q^{91} +3.38637e8 q^{92} +(-6.59798e8 - 1.14280e9i) q^{94} +(2.71598e8 + 4.70422e8i) q^{95} +3.75562e8 q^{97} +(1.04489e9 - 7.05074e8i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 18 q^{2} - 940 q^{4} - 1533 q^{5} - 1036 q^{7} - 34272 q^{8} + 4298 q^{10} - 42213 q^{11} - 319676 q^{13} + 39522 q^{14} + 322064 q^{16} - 324681 q^{17} - 16121 q^{19} + 350616 q^{20} - 62692 q^{22}+ \cdots + 2404833858 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −15.6185 + 27.0520i −0.690246 + 1.19554i 0.281511 + 0.959558i \(0.409164\pi\)
−0.971757 + 0.235983i \(0.924169\pi\)
\(3\) 0 0
\(4\) −231.874 − 401.617i −0.452878 − 0.784408i
\(5\) 239.755 − 415.269i 0.171555 − 0.297142i −0.767409 − 0.641158i \(-0.778454\pi\)
0.938964 + 0.344016i \(0.111788\pi\)
\(6\) 0 0
\(7\) 1428.40 − 6189.77i 0.224859 − 0.974391i
\(8\) −1507.26 −0.130102
\(9\) 0 0
\(10\) 7489.23 + 12971.7i 0.236830 + 0.410202i
\(11\) −33637.5 − 58261.9i −0.692719 − 1.19982i −0.970944 − 0.239309i \(-0.923079\pi\)
0.278224 − 0.960516i \(-0.410254\pi\)
\(12\) 0 0
\(13\) 39620.0 0.384742 0.192371 − 0.981322i \(-0.438382\pi\)
0.192371 + 0.981322i \(0.438382\pi\)
\(14\) 145136. + 135316.i 1.00972 + 0.941397i
\(15\) 0 0
\(16\) 142260. − 246402.i 0.542681 − 0.939951i
\(17\) 106607. + 184649.i 0.309575 + 0.536199i 0.978269 − 0.207338i \(-0.0664800\pi\)
−0.668695 + 0.743537i \(0.733147\pi\)
\(18\) 0 0
\(19\) −566407. + 981046.i −0.997097 + 1.72702i −0.432612 + 0.901580i \(0.642408\pi\)
−0.564485 + 0.825443i \(0.690925\pi\)
\(20\) −222372. −0.310774
\(21\) 0 0
\(22\) 2.10147e6 1.91259
\(23\) −365109. + 632388.i −0.272049 + 0.471203i −0.969386 − 0.245540i \(-0.921035\pi\)
0.697337 + 0.716743i \(0.254368\pi\)
\(24\) 0 0
\(25\) 861597. + 1.49233e6i 0.441138 + 0.764073i
\(26\) −618804. + 1.07180e6i −0.265566 + 0.459974i
\(27\) 0 0
\(28\) −2.81713e6 + 861574.i −0.866154 + 0.264900i
\(29\) −383457. −0.100676 −0.0503379 − 0.998732i \(-0.516030\pi\)
−0.0503379 + 0.998732i \(0.516030\pi\)
\(30\) 0 0
\(31\) 2.73393e6 + 4.73530e6i 0.531691 + 0.920915i 0.999316 + 0.0369885i \(0.0117765\pi\)
−0.467625 + 0.883927i \(0.654890\pi\)
\(32\) 4.05793e6 + 7.02853e6i 0.684115 + 1.18492i
\(33\) 0 0
\(34\) −6.66015e6 −0.854730
\(35\) −2.22795e6 − 2.07720e6i −0.250957 − 0.233977i
\(36\) 0 0
\(37\) 3.83476e6 − 6.64200e6i 0.336380 − 0.582627i −0.647369 − 0.762177i \(-0.724131\pi\)
0.983749 + 0.179549i \(0.0574640\pi\)
\(38\) −1.76928e7 − 3.06449e7i −1.37648 − 2.38414i
\(39\) 0 0
\(40\) −361374. + 625919.i −0.0223196 + 0.0386588i
\(41\) −1.08865e6 −0.0601671 −0.0300836 − 0.999547i \(-0.509577\pi\)
−0.0300836 + 0.999547i \(0.509577\pi\)
\(42\) 0 0
\(43\) −1.92137e7 −0.857044 −0.428522 − 0.903531i \(-0.640965\pi\)
−0.428522 + 0.903531i \(0.640965\pi\)
\(44\) −1.55993e7 + 2.70188e7i −0.627435 + 1.08675i
\(45\) 0 0
\(46\) −1.14049e7 − 1.97539e7i −0.375562 − 0.650492i
\(47\) −2.11223e7 + 3.65850e7i −0.631396 + 1.09361i 0.355871 + 0.934535i \(0.384184\pi\)
−0.987267 + 0.159074i \(0.949149\pi\)
\(48\) 0 0
\(49\) −3.62729e7 − 1.76830e7i −0.898877 − 0.438201i
\(50\) −5.38273e7 −1.21797
\(51\) 0 0
\(52\) −9.18684e6 − 1.59121e7i −0.174241 − 0.301795i
\(53\) −2.68028e7 − 4.64237e7i −0.466593 − 0.808163i 0.532679 − 0.846317i \(-0.321185\pi\)
−0.999272 + 0.0381548i \(0.987852\pi\)
\(54\) 0 0
\(55\) −3.22591e7 −0.475358
\(56\) −2.15298e6 + 9.32961e6i −0.0292545 + 0.126770i
\(57\) 0 0
\(58\) 5.98901e6 − 1.03733e7i 0.0694910 − 0.120362i
\(59\) 9.06495e6 + 1.57010e7i 0.0973938 + 0.168691i 0.910605 − 0.413277i \(-0.135616\pi\)
−0.813211 + 0.581968i \(0.802283\pi\)
\(60\) 0 0
\(61\) −1.48303e7 + 2.56868e7i −0.137140 + 0.237534i −0.926413 − 0.376509i \(-0.877125\pi\)
0.789273 + 0.614043i \(0.210458\pi\)
\(62\) −1.70799e8 −1.46799
\(63\) 0 0
\(64\) −1.07840e8 −0.803469
\(65\) 9.49911e6 − 1.64529e7i 0.0660044 − 0.114323i
\(66\) 0 0
\(67\) 8.34811e7 + 1.44594e8i 0.506118 + 0.876622i 0.999975 + 0.00707865i \(0.00225322\pi\)
−0.493857 + 0.869543i \(0.664413\pi\)
\(68\) 4.94387e7 − 8.56303e7i 0.280399 − 0.485666i
\(69\) 0 0
\(70\) 9.09897e7 − 2.78278e7i 0.452951 − 0.138528i
\(71\) 3.15617e8 1.47400 0.737001 − 0.675892i \(-0.236241\pi\)
0.737001 + 0.675892i \(0.236241\pi\)
\(72\) 0 0
\(73\) 1.51081e7 + 2.61679e7i 0.0622667 + 0.107849i 0.895478 − 0.445105i \(-0.146834\pi\)
−0.833212 + 0.552954i \(0.813500\pi\)
\(74\) 1.19786e8 + 2.07476e8i 0.464370 + 0.804312i
\(75\) 0 0
\(76\) 5.25340e8 1.80626
\(77\) −4.08676e8 + 1.24987e8i −1.32486 + 0.405189i
\(78\) 0 0
\(79\) −2.62085e8 + 4.53944e8i −0.757041 + 1.31123i 0.187312 + 0.982301i \(0.440023\pi\)
−0.944353 + 0.328934i \(0.893311\pi\)
\(80\) −6.82155e7 − 1.18153e8i −0.186199 − 0.322507i
\(81\) 0 0
\(82\) 1.70030e7 − 2.94500e7i 0.0415301 − 0.0719323i
\(83\) 6.01000e8 1.39003 0.695013 − 0.718997i \(-0.255399\pi\)
0.695013 + 0.718997i \(0.255399\pi\)
\(84\) 0 0
\(85\) 1.02238e8 0.212436
\(86\) 3.00089e8 − 5.19769e8i 0.591571 − 1.02463i
\(87\) 0 0
\(88\) 5.07006e7 + 8.78160e7i 0.0901241 + 0.156100i
\(89\) −4.80628e8 + 8.32473e8i −0.811997 + 1.40642i 0.0994677 + 0.995041i \(0.468286\pi\)
−0.911464 + 0.411379i \(0.865047\pi\)
\(90\) 0 0
\(91\) 5.65933e7 − 2.45239e8i 0.0865125 − 0.374889i
\(92\) 3.38637e8 0.492821
\(93\) 0 0
\(94\) −6.59798e8 − 1.14280e9i −0.871636 − 1.50972i
\(95\) 2.71598e8 + 4.70422e8i 0.342114 + 0.592559i
\(96\) 0 0
\(97\) 3.75562e8 0.430734 0.215367 − 0.976533i \(-0.430905\pi\)
0.215367 + 0.976533i \(0.430905\pi\)
\(98\) 1.04489e9 − 7.05074e8i 1.14433 − 0.772178i
\(99\) 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 63.10.e.b.37.1 10
3.2 odd 2 7.10.c.a.2.5 ✓ 10
7.4 even 3 inner 63.10.e.b.46.1 10
12.11 even 2 112.10.i.c.65.4 10
21.2 odd 6 49.10.a.e.1.1 5
21.5 even 6 49.10.a.f.1.1 5
21.11 odd 6 7.10.c.a.4.5 yes 10
21.17 even 6 49.10.c.g.18.5 10
21.20 even 2 49.10.c.g.30.5 10
84.11 even 6 112.10.i.c.81.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.c.a.2.5 ✓ 10 3.2 odd 2
7.10.c.a.4.5 yes 10 21.11 odd 6
49.10.a.e.1.1 5 21.2 odd 6
49.10.a.f.1.1 5 21.5 even 6
49.10.c.g.18.5 10 21.17 even 6
49.10.c.g.30.5 10 21.20 even 2
63.10.e.b.37.1 10 1.1 even 1 trivial
63.10.e.b.46.1 10 7.4 even 3 inner
112.10.i.c.65.4 10 12.11 even 2
112.10.i.c.81.4 10 84.11 even 6