Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-3] 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: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-83})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 20x^{2} - 21x + 441 \) 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 46.1
Root \(4.19493 + 1.84460i\) of defining polynomial
Character \(\chi\) \(=\) 63.46
Dual form 63.6.e.c.37.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.69493 - 8.13186i) q^{2} +(-28.0848 + 48.6443i) q^{4} +(-35.8645 - 62.1192i) q^{5} +(-87.5000 - 95.6596i) q^{7} +226.949 q^{8} +(-336.763 + 583.291i) q^{10} +(-280.305 + 485.503i) q^{11} +533.509 q^{13} +(-367.084 + 1160.65i) q^{14} +(-166.798 - 288.903i) q^{16} +(502.850 - 870.962i) q^{17} +(-684.263 - 1185.18i) q^{19} +4028.99 q^{20} +5264.05 q^{22} +(1614.04 + 2795.60i) q^{23} +(-1010.03 + 1749.42i) q^{25} +(-2504.79 - 4338.42i) q^{26} +(7110.71 - 1569.80i) q^{28} +753.456 q^{29} +(-4103.21 + 7106.97i) q^{31} +(2064.97 - 3576.64i) q^{32} -9443.39 q^{34} +(-2804.15 + 8866.21i) q^{35} +(1404.33 + 2432.37i) q^{37} +(-6425.14 + 11128.7i) q^{38} +(-8139.43 - 14097.9i) q^{40} -245.827 q^{41} -17504.5 q^{43} +(-15744.6 - 27270.5i) q^{44} +(15155.6 - 26250.3i) q^{46} +(-8172.74 - 14155.6i) q^{47} +(-1494.50 + 16740.4i) q^{49} +18968.1 q^{50} +(-14983.5 + 25952.2i) q^{52} +(-14820.8 + 25670.4i) q^{53} +40212.0 q^{55} +(-19858.1 - 21709.9i) q^{56} +(-3537.43 - 6127.00i) q^{58} +(-5178.05 + 8968.65i) q^{59} +(-477.089 - 826.343i) q^{61} +77057.2 q^{62} -49454.8 q^{64} +(-19134.0 - 33141.1i) q^{65} +(9907.60 - 17160.5i) q^{67} +(28244.9 + 48921.6i) q^{68} +(85264.1 - 18823.3i) q^{70} -62125.4 q^{71} +(-13554.8 + 23477.6i) q^{73} +(13186.5 - 22839.7i) q^{74} +76869.6 q^{76} +(70969.7 - 15667.6i) q^{77} +(-22343.7 - 38700.4i) q^{79} +(-11964.3 + 20722.8i) q^{80} +(1154.14 + 1999.03i) q^{82} -15606.6 q^{83} -72137.9 q^{85} +(82182.5 + 142344. i) q^{86} +(-63615.0 + 110184. i) q^{88} +(6817.66 + 11808.5i) q^{89} +(-46682.0 - 51035.2i) q^{91} -181320. q^{92} +(-76740.9 + 132919. i) q^{94} +(-49081.6 + 85011.8i) q^{95} -12919.5 q^{97} +(143147. - 66442.1i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 65 q^{4} - 33 q^{5} - 350 q^{7} + 750 q^{8} - 921 q^{10} - 1137 q^{11} + 1850 q^{13} - 2352 q^{14} + 895 q^{16} - 324 q^{17} - 2311 q^{19} + 7374 q^{20} + 3162 q^{22} + 1596 q^{23} - 395 q^{25}+ \cdots + 462021 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.69493 − 8.13186i −0.829955 − 1.43752i −0.898073 − 0.439847i \(-0.855033\pi\)
0.0681181 − 0.997677i \(-0.478301\pi\)
\(3\) 0 0
\(4\) −28.0848 + 48.6443i −0.877650 + 1.52013i
\(5\) −35.8645 − 62.1192i −0.641564 − 1.11122i −0.985084 − 0.172076i \(-0.944952\pi\)
0.343519 − 0.939146i \(-0.388381\pi\)
\(6\) 0 0
\(7\) −87.5000 − 95.6596i −0.674937 − 0.737876i
\(8\) 226.949 1.25373
\(9\) 0 0
\(10\) −336.763 + 583.291i −1.06494 + 1.84453i
\(11\) −280.305 + 485.503i −0.698472 + 1.20979i 0.270524 + 0.962713i \(0.412803\pi\)
−0.968996 + 0.247076i \(0.920530\pi\)
\(12\) 0 0
\(13\) 533.509 0.875555 0.437777 − 0.899083i \(-0.355766\pi\)
0.437777 + 0.899083i \(0.355766\pi\)
\(14\) −367.084 + 1160.65i −0.500547 + 1.58264i
\(15\) 0 0
\(16\) −166.798 − 288.903i −0.162889 − 0.282132i
\(17\) 502.850 − 870.962i 0.422004 − 0.730932i −0.574132 − 0.818763i \(-0.694660\pi\)
0.996135 + 0.0878311i \(0.0279936\pi\)
\(18\) 0 0
\(19\) −684.263 − 1185.18i −0.434850 − 0.753182i 0.562434 − 0.826842i \(-0.309865\pi\)
−0.997283 + 0.0736606i \(0.976532\pi\)
\(20\) 4028.99 2.25228
\(21\) 0 0
\(22\) 5264.05 2.31880
\(23\) 1614.04 + 2795.60i 0.636201 + 1.10193i 0.986259 + 0.165205i \(0.0528286\pi\)
−0.350058 + 0.936728i \(0.613838\pi\)
\(24\) 0 0
\(25\) −1010.03 + 1749.42i −0.323209 + 0.559815i
\(26\) −2504.79 − 4338.42i −0.726671 − 1.25863i
\(27\) 0 0
\(28\) 7110.71 − 1569.80i 1.71403 − 0.378398i
\(29\) 753.456 0.166365 0.0831827 − 0.996534i \(-0.473492\pi\)
0.0831827 + 0.996534i \(0.473492\pi\)
\(30\) 0 0
\(31\) −4103.21 + 7106.97i −0.766866 + 1.32825i 0.172389 + 0.985029i \(0.444852\pi\)
−0.939254 + 0.343222i \(0.888482\pi\)
\(32\) 2064.97 − 3576.64i 0.356484 − 0.617448i
\(33\) 0 0
\(34\) −9443.39 −1.40098
\(35\) −2804.15 + 8866.21i −0.386929 + 1.22340i
\(36\) 0 0
\(37\) 1404.33 + 2432.37i 0.168642 + 0.292096i 0.937943 − 0.346791i \(-0.112729\pi\)
−0.769301 + 0.638887i \(0.779395\pi\)
\(38\) −6425.14 + 11128.7i −0.721811 + 1.25021i
\(39\) 0 0
\(40\) −8139.43 − 14097.9i −0.804348 − 1.39317i
\(41\) −245.827 −0.0228387 −0.0114193 − 0.999935i \(-0.503635\pi\)
−0.0114193 + 0.999935i \(0.503635\pi\)
\(42\) 0 0
\(43\) −17504.5 −1.44371 −0.721853 − 0.692047i \(-0.756709\pi\)
−0.721853 + 0.692047i \(0.756709\pi\)
\(44\) −15744.6 − 27270.5i −1.22603 − 2.12354i
\(45\) 0 0
\(46\) 15155.6 − 26250.3i 1.05604 − 1.82911i
\(47\) −8172.74 − 14155.6i −0.539663 − 0.934725i −0.998922 − 0.0464219i \(-0.985218\pi\)
0.459258 − 0.888303i \(-0.348115\pi\)
\(48\) 0 0
\(49\) −1494.50 + 16740.4i −0.0889213 + 0.996039i
\(50\) 18968.1 1.07300
\(51\) 0 0
\(52\) −14983.5 + 25952.2i −0.768430 + 1.33096i
\(53\) −14820.8 + 25670.4i −0.724741 + 1.25529i 0.234340 + 0.972155i \(0.424707\pi\)
−0.959081 + 0.283133i \(0.908626\pi\)
\(54\) 0 0
\(55\) 40212.0 1.79246
\(56\) −19858.1 − 21709.9i −0.846188 − 0.925097i
\(57\) 0 0
\(58\) −3537.43 − 6127.00i −0.138076 − 0.239154i
\(59\) −5178.05 + 8968.65i −0.193659 + 0.335426i −0.946460 − 0.322821i \(-0.895369\pi\)
0.752801 + 0.658248i \(0.228702\pi\)
\(60\) 0 0
\(61\) −477.089 − 826.343i −0.0164163 − 0.0284339i 0.857701 − 0.514150i \(-0.171892\pi\)
−0.874117 + 0.485716i \(0.838559\pi\)
\(62\) 77057.2 2.54586
\(63\) 0 0
\(64\) −49454.8 −1.50924
\(65\) −19134.0 − 33141.1i −0.561725 − 0.972935i
\(66\) 0 0
\(67\) 9907.60 − 17160.5i 0.269638 − 0.467027i −0.699130 − 0.714994i \(-0.746429\pi\)
0.968768 + 0.247967i \(0.0797626\pi\)
\(68\) 28244.9 + 48921.6i 0.740743 + 1.28300i
\(69\) 0 0
\(70\) 85264.1 − 18823.3i 2.07980 − 0.459147i
\(71\) −62125.4 −1.46259 −0.731296 − 0.682060i \(-0.761084\pi\)
−0.731296 + 0.682060i \(0.761084\pi\)
\(72\) 0 0
\(73\) −13554.8 + 23477.6i −0.297705 + 0.515641i −0.975611 − 0.219509i \(-0.929555\pi\)
0.677905 + 0.735149i \(0.262888\pi\)
\(74\) 13186.5 − 22839.7i 0.279930 − 0.484853i
\(75\) 0 0
\(76\) 76869.6 1.52658
\(77\) 70969.7 − 15667.6i 1.36410 − 0.301145i
\(78\) 0 0
\(79\) −22343.7 − 38700.4i −0.402798 − 0.697666i 0.591265 − 0.806477i \(-0.298629\pi\)
−0.994062 + 0.108812i \(0.965295\pi\)
\(80\) −11964.3 + 20722.8i −0.209008 + 0.362012i
\(81\) 0 0
\(82\) 1154.14 + 1999.03i 0.0189551 + 0.0328311i
\(83\) −15606.6 −0.248665 −0.124332 − 0.992241i \(-0.539679\pi\)
−0.124332 + 0.992241i \(0.539679\pi\)
\(84\) 0 0
\(85\) −72137.9 −1.08297
\(86\) 82182.5 + 142344.i 1.19821 + 2.07536i
\(87\) 0 0
\(88\) −63615.0 + 110184.i −0.875696 + 1.51675i
\(89\) 6817.66 + 11808.5i 0.0912347 + 0.158023i 0.908031 − 0.418903i \(-0.137585\pi\)
−0.816796 + 0.576926i \(0.804252\pi\)
\(90\) 0 0
\(91\) −46682.0 − 51035.2i −0.590944 − 0.646050i
\(92\) −181320. −2.23345
\(93\) 0 0
\(94\) −76740.9 + 132919.i −0.895793 + 1.55156i
\(95\) −49081.6 + 85011.8i −0.557968 + 0.966429i
\(96\) 0 0
\(97\) −12919.5 −0.139417 −0.0697086 − 0.997567i \(-0.522207\pi\)
−0.0697086 + 0.997567i \(0.522207\pi\)
\(98\) 143147. − 66442.1i 1.50563 − 0.698841i
\(99\) 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 63.6.e.c.46.1 4
3.2 odd 2 21.6.e.b.4.2 ✓ 4
7.2 even 3 inner 63.6.e.c.37.1 4
7.3 odd 6 441.6.a.s.1.2 2
7.4 even 3 441.6.a.t.1.2 2
12.11 even 2 336.6.q.e.193.2 4
21.2 odd 6 21.6.e.b.16.2 yes 4
21.5 even 6 147.6.e.l.79.2 4
21.11 odd 6 147.6.a.i.1.1 2
21.17 even 6 147.6.a.k.1.1 2
21.20 even 2 147.6.e.l.67.2 4
84.23 even 6 336.6.q.e.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.2 ✓ 4 3.2 odd 2
21.6.e.b.16.2 yes 4 21.2 odd 6
63.6.e.c.37.1 4 7.2 even 3 inner
63.6.e.c.46.1 4 1.1 even 1 trivial
147.6.a.i.1.1 2 21.11 odd 6
147.6.a.k.1.1 2 21.17 even 6
147.6.e.l.67.2 4 21.20 even 2
147.6.e.l.79.2 4 21.5 even 6
336.6.q.e.193.2 4 12.11 even 2
336.6.q.e.289.2 4 84.23 even 6
441.6.a.s.1.2 2 7.3 odd 6
441.6.a.t.1.2 2 7.4 even 3