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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [639,2,Mod(10,639)] 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("639.10"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(639, base_ring=CyclotomicField(70)) chi = DirichletCharacter(H, H._module([0, 34])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.v (of order \(35\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,22,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.10244068916\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(5\) over \(\Q(\zeta_{35})\)
Twist minimal: no (minimal twist has level 71)
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

Embedding invariants

Embedding label 10.1
Character \(\chi\) \(=\) 639.10
Dual form 639.2.v.a.64.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.59197 + 0.439356i) q^{2} +(0.624443 - 0.373087i) q^{4} +(-0.346904 + 1.06766i) q^{5} +(-0.160876 - 3.58218i) q^{7} +(1.45238 - 1.51908i) q^{8} +(0.0831780 - 1.85210i) q^{10} +(2.39818 + 0.435205i) q^{11} +(-1.60616 + 0.291475i) q^{13} +(1.82996 + 5.63204i) q^{14} +(-2.33413 + 4.33754i) q^{16} +(-4.23960 + 3.08025i) q^{17} +(-2.55135 + 2.22905i) q^{19} +(0.181709 + 0.796119i) q^{20} +(-4.00904 + 0.360820i) q^{22} +(-4.87878 - 6.11780i) q^{23} +(3.02553 + 2.19817i) q^{25} +(2.42890 - 1.16970i) q^{26} +(-1.43692 - 2.17684i) q^{28} +(-2.50916 - 0.225828i) q^{29} +(-3.82972 - 7.11681i) q^{31} +(0.874814 - 3.83281i) q^{32} +(5.39600 - 6.76637i) q^{34} +(3.88036 + 1.07091i) q^{35} +(1.00279 - 1.25746i) q^{37} +(3.08233 - 4.66954i) q^{38} +(1.11802 + 2.07763i) q^{40} +(3.61060 + 1.73877i) q^{41} +(-3.20849 - 4.86065i) q^{43} +(1.65989 - 0.622969i) q^{44} +(10.4548 + 7.59584i) q^{46} +(-4.97200 - 11.6326i) q^{47} +(-5.83429 + 0.525096i) q^{49} +(-5.78233 - 2.17015i) q^{50} +(-0.894210 + 0.781248i) q^{52} +(-2.58438 - 1.54409i) q^{53} +(-1.29659 + 2.40947i) q^{55} +(-5.67525 - 4.95832i) q^{56} +(4.09373 - 0.742903i) q^{58} +(1.57866 + 11.6541i) q^{59} +(0.413377 - 9.20456i) q^{61} +(9.22362 + 9.64715i) q^{62} +(-0.150690 - 3.35538i) q^{64} +(0.245987 - 1.81595i) q^{65} +(-9.52606 + 5.69156i) q^{67} +(-1.49819 + 3.50518i) q^{68} -6.64794 q^{70} +(-5.96969 - 5.94667i) q^{71} +(-8.95105 + 2.47033i) q^{73} +(-1.04394 + 2.44243i) q^{74} +(-0.761543 + 2.34379i) q^{76} +(1.17317 - 8.66071i) q^{77} +(8.14963 - 8.52384i) q^{79} +(-3.82131 - 3.99678i) q^{80} +(-6.51192 - 1.18174i) q^{82} +(-0.347401 - 2.56462i) q^{83} +(-1.81793 - 5.59501i) q^{85} +(7.24338 + 6.32835i) q^{86} +(4.14419 - 3.01093i) q^{88} +(1.43761 + 0.858930i) q^{89} +(1.30251 + 5.70667i) q^{91} +(-5.32899 - 2.00000i) q^{92} +(13.0261 + 16.3342i) q^{94} +(-1.49480 - 3.49725i) q^{95} +(5.45768 - 2.62828i) q^{97} +(9.05733 - 3.39927i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 22 q^{2} - 18 q^{4} + 20 q^{5} - 27 q^{7} + 27 q^{8} - 8 q^{10} + 27 q^{11} - 31 q^{13} - 2 q^{14} + 30 q^{16} - 9 q^{17} - 31 q^{19} - 72 q^{20} - 24 q^{22} + 6 q^{23} - 42 q^{25} - 77 q^{26} - 18 q^{28}+ \cdots - 128 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/639\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(569\)
\(\chi(n)\) \(e\left(\frac{17}{35}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.59197 + 0.439356i −1.12569 + 0.310672i −0.778825 − 0.627242i \(-0.784184\pi\)
−0.346869 + 0.937913i \(0.612755\pi\)
\(3\) 0 0
\(4\) 0.624443 − 0.373087i 0.312221 − 0.186544i
\(5\) −0.346904 + 1.06766i −0.155140 + 0.477473i −0.998175 − 0.0603863i \(-0.980767\pi\)
0.843035 + 0.537859i \(0.180767\pi\)
\(6\) 0 0
\(7\) −0.160876 − 3.58218i −0.0608053 − 1.35394i −0.765047 − 0.643975i \(-0.777284\pi\)
0.704241 − 0.709961i \(-0.251287\pi\)
\(8\) 1.45238 − 1.51908i 0.513496 − 0.537074i
\(9\) 0 0
\(10\) 0.0831780 − 1.85210i 0.0263032 − 0.585686i
\(11\) 2.39818 + 0.435205i 0.723078 + 0.131219i 0.527605 − 0.849490i \(-0.323090\pi\)
0.195473 + 0.980709i \(0.437376\pi\)
\(12\) 0 0
\(13\) −1.60616 + 0.291475i −0.445469 + 0.0808407i −0.396656 − 0.917967i \(-0.629830\pi\)
−0.0488129 + 0.998808i \(0.515544\pi\)
\(14\) 1.82996 + 5.63204i 0.489078 + 1.50523i
\(15\) 0 0
\(16\) −2.33413 + 4.33754i −0.583533 + 1.08439i
\(17\) −4.23960 + 3.08025i −1.02825 + 0.747070i −0.967959 − 0.251110i \(-0.919204\pi\)
−0.0602958 + 0.998181i \(0.519204\pi\)
\(18\) 0 0
\(19\) −2.55135 + 2.22905i −0.585320 + 0.511379i −0.898883 − 0.438189i \(-0.855620\pi\)
0.313563 + 0.949567i \(0.398477\pi\)
\(20\) 0.181709 + 0.796119i 0.0406314 + 0.178018i
\(21\) 0 0
\(22\) −4.00904 + 0.360820i −0.854731 + 0.0769272i
\(23\) −4.87878 − 6.11780i −1.01730 − 1.27565i −0.960799 − 0.277246i \(-0.910578\pi\)
−0.0564973 − 0.998403i \(-0.517993\pi\)
\(24\) 0 0
\(25\) 3.02553 + 2.19817i 0.605105 + 0.439635i
\(26\) 2.42890 − 1.16970i 0.476347 − 0.229397i
\(27\) 0 0
\(28\) −1.43692 − 2.17684i −0.271553 − 0.411385i
\(29\) −2.50916 − 0.225828i −0.465939 − 0.0419353i −0.145817 − 0.989312i \(-0.546581\pi\)
−0.320122 + 0.947376i \(0.603724\pi\)
\(30\) 0 0
\(31\) −3.82972 − 7.11681i −0.687838 − 1.27822i −0.949715 − 0.313115i \(-0.898627\pi\)
0.261877 − 0.965101i \(-0.415658\pi\)
\(32\) 0.874814 − 3.83281i 0.154647 − 0.677552i
\(33\) 0 0
\(34\) 5.39600 − 6.76637i 0.925406 − 1.16042i
\(35\) 3.88036 + 1.07091i 0.655901 + 0.181017i
\(36\) 0 0
\(37\) 1.00279 − 1.25746i 0.164858 − 0.206726i −0.692540 − 0.721380i \(-0.743508\pi\)
0.857398 + 0.514654i \(0.172080\pi\)
\(38\) 3.08233 − 4.66954i 0.500021 − 0.757499i
\(39\) 0 0
\(40\) 1.11802 + 2.07763i 0.176774 + 0.328502i
\(41\) 3.61060 + 1.73877i 0.563881 + 0.271551i 0.694033 − 0.719943i \(-0.255832\pi\)
−0.130152 + 0.991494i \(0.541546\pi\)
\(42\) 0 0
\(43\) −3.20849 − 4.86065i −0.489290 − 0.741242i 0.503120 − 0.864217i \(-0.332185\pi\)
−0.992410 + 0.122974i \(0.960757\pi\)
\(44\) 1.65989 − 0.622969i 0.250238 − 0.0939161i
\(45\) 0 0
\(46\) 10.4548 + 7.59584i 1.54147 + 1.11995i
\(47\) −4.97200 − 11.6326i −0.725241 − 1.69679i −0.717249 − 0.696817i \(-0.754599\pi\)
−0.00799209 − 0.999968i \(-0.502544\pi\)
\(48\) 0 0
\(49\) −5.83429 + 0.525096i −0.833470 + 0.0750137i
\(50\) −5.78233 − 2.17015i −0.817745 − 0.306905i
\(51\) 0 0
\(52\) −0.894210 + 0.781248i −0.124005 + 0.108340i
\(53\) −2.58438 − 1.54409i −0.354992 − 0.212098i 0.324320 − 0.945947i \(-0.394864\pi\)
−0.679312 + 0.733850i \(0.737722\pi\)
\(54\) 0 0
\(55\) −1.29659 + 2.40947i −0.174832 + 0.324893i
\(56\) −5.67525 − 4.95832i −0.758387 − 0.662583i
\(57\) 0 0
\(58\) 4.09373 − 0.742903i 0.537533 − 0.0975479i
\(59\) 1.57866 + 11.6541i 0.205524 + 1.51724i 0.740318 + 0.672257i \(0.234675\pi\)
−0.534794 + 0.844982i \(0.679611\pi\)
\(60\) 0 0
\(61\) 0.413377 − 9.20456i 0.0529275 − 1.17852i −0.780232 − 0.625490i \(-0.784899\pi\)
0.833160 − 0.553033i \(-0.186530\pi\)
\(62\) 9.22362 + 9.64715i 1.17140 + 1.22519i
\(63\) 0 0
\(64\) −0.150690 − 3.35538i −0.0188363 − 0.419423i
\(65\) 0.245987 − 1.81595i 0.0305110 − 0.225241i
\(66\) 0 0
\(67\) −9.52606 + 5.69156i −1.16379 + 0.695334i −0.958982 − 0.283467i \(-0.908515\pi\)
−0.204811 + 0.978801i \(0.565658\pi\)
\(68\) −1.49819 + 3.50518i −0.181682 + 0.425065i
\(69\) 0 0
\(70\) −6.64794 −0.794581
\(71\) −5.96969 − 5.94667i −0.708472 − 0.705739i
\(72\) 0 0
\(73\) −8.95105 + 2.47033i −1.04764 + 0.289131i −0.747151 − 0.664654i \(-0.768579\pi\)
−0.300490 + 0.953785i \(0.597150\pi\)
\(74\) −1.04394 + 2.44243i −0.121356 + 0.283927i
\(75\) 0 0
\(76\) −0.761543 + 2.34379i −0.0873550 + 0.268851i
\(77\) 1.17317 − 8.66071i 0.133696 − 0.986980i
\(78\) 0 0
\(79\) 8.14963 − 8.52384i 0.916905 − 0.959007i −0.0823884 − 0.996600i \(-0.526255\pi\)
0.999293 + 0.0375932i \(0.0119691\pi\)
\(80\) −3.82131 − 3.99678i −0.427235 − 0.446853i
\(81\) 0 0
\(82\) −6.51192 − 1.18174i −0.719121 − 0.130501i
\(83\) −0.347401 − 2.56462i −0.0381322 − 0.281503i −0.999969 − 0.00787935i \(-0.997492\pi\)
0.961837 − 0.273624i \(-0.0882224\pi\)
\(84\) 0 0
\(85\) −1.81793 − 5.59501i −0.197182 − 0.606864i
\(86\) 7.24338 + 6.32835i 0.781074 + 0.682404i
\(87\) 0 0
\(88\) 4.14419 − 3.01093i 0.441772 − 0.320966i
\(89\) 1.43761 + 0.858930i 0.152386 + 0.0910464i 0.587068 − 0.809538i \(-0.300282\pi\)
−0.434682 + 0.900584i \(0.643139\pi\)
\(90\) 0 0
\(91\) 1.30251 + 5.70667i 0.136540 + 0.598221i
\(92\) −5.32899 − 2.00000i −0.555586 − 0.208515i
\(93\) 0 0
\(94\) 13.0261 + 16.3342i 1.34354 + 1.68475i
\(95\) −1.49480 − 3.49725i −0.153363 − 0.358810i
\(96\) 0 0
\(97\) 5.45768 − 2.62828i 0.554144 − 0.266862i −0.135786 − 0.990738i \(-0.543356\pi\)
0.689929 + 0.723877i \(0.257642\pi\)
\(98\) 9.05733 − 3.39927i 0.914928 − 0.343378i
\(99\) 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 639.2.v.a.10.1 120
3.2 odd 2 71.2.g.a.10.5 ✓ 120
71.64 even 35 inner 639.2.v.a.64.1 120
213.8 odd 70 5041.2.a.s.1.15 60
213.134 even 70 5041.2.a.t.1.15 60
213.206 odd 70 71.2.g.a.64.5 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
71.2.g.a.10.5 ✓ 120 3.2 odd 2
71.2.g.a.64.5 yes 120 213.206 odd 70
639.2.v.a.10.1 120 1.1 even 1 trivial
639.2.v.a.64.1 120 71.64 even 35 inner
5041.2.a.s.1.15 60 213.8 odd 70
5041.2.a.t.1.15 60 213.134 even 70