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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [207,3,Mod(10,207)] 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("207.10"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 207.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,11] 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: \(5.64034147226\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 23)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 145.1
Character \(\chi\) \(=\) 207.145
Dual form 207.3.j.a.10.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.59301 + 1.83844i) q^{2} +(-0.272894 - 1.89802i) q^{4} +(2.69128 - 1.22907i) q^{5} +(-1.33225 - 4.53722i) q^{7} +(-4.26162 - 2.73877i) q^{8} +(-2.02769 + 6.90566i) q^{10} +(-11.0247 + 9.55293i) q^{11} +(-22.7531 - 6.68090i) q^{13} +(10.4637 + 4.77860i) q^{14} +(19.1833 - 5.63274i) q^{16} +(-12.1869 - 1.75221i) q^{17} +(5.54565 - 0.797344i) q^{19} +(-3.06723 - 4.77270i) q^{20} -35.4861i q^{22} +(-5.15862 - 22.4140i) q^{23} +(-10.6391 + 12.2782i) q^{25} +(48.5283 - 31.1873i) q^{26} +(-8.24818 + 3.76682i) q^{28} +(3.08261 - 21.4400i) q^{29} +(-0.0511435 - 0.0328679i) q^{31} +(-11.7863 + 25.8083i) q^{32} +(22.6352 - 19.6135i) q^{34} +(-9.16200 - 10.5735i) q^{35} +(13.3445 + 6.09421i) q^{37} +(-7.36843 + 11.4655i) q^{38} +(-14.8353 - 2.13300i) q^{40} +(-4.95227 - 10.8440i) q^{41} +(-27.4308 - 42.6832i) q^{43} +(21.1402 + 18.3181i) q^{44} +(49.4245 + 26.2221i) q^{46} +23.0534 q^{47} +(22.4099 - 14.4020i) q^{49} +(-5.62443 - 39.1188i) q^{50} +(-6.47131 + 45.0090i) q^{52} +(28.2602 + 96.2454i) q^{53} +(-17.9293 + 39.2596i) q^{55} +(-6.74889 + 22.9846i) q^{56} +(34.5055 + 39.8214i) q^{58} +(-34.0311 - 9.99245i) q^{59} +(-49.9475 + 77.7198i) q^{61} +(0.141898 - 0.0416649i) q^{62} +(4.55063 + 9.96449i) q^{64} +(-69.4461 + 9.98484i) q^{65} +(-72.5767 - 62.8880i) q^{67} +23.6092i q^{68} +34.0339 q^{70} +(-35.6457 + 41.1374i) q^{71} +(6.81309 + 47.3861i) q^{73} +(-32.4617 + 14.8248i) q^{74} +(-3.02675 - 10.3082i) q^{76} +(58.0313 + 37.2945i) q^{77} +(-10.2622 + 34.9499i) q^{79} +(44.7047 - 38.7368i) q^{80} +(27.8249 + 8.17014i) q^{82} +(-95.0493 - 43.4076i) q^{83} +(-34.9519 + 10.2628i) q^{85} +(122.168 + 17.5651i) q^{86} +(73.1462 - 10.5168i) q^{88} +(2.45164 + 3.81483i) q^{89} +112.136i q^{91} +(-41.1346 + 15.9078i) q^{92} +(-36.7244 + 42.3822i) q^{94} +(13.9449 - 8.96184i) q^{95} +(119.559 - 54.6009i) q^{97} +(-9.22220 + 64.1418i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 11 q^{2} - 23 q^{4} + 11 q^{5} - 11 q^{7} - 10 q^{8} - 11 q^{10} + 11 q^{11} - 11 q^{13} + 11 q^{14} + 73 q^{16} - 44 q^{17} + 22 q^{19} - 77 q^{20} - 36 q^{23} - 152 q^{25} + 186 q^{26} - 275 q^{28}+ \cdots - 77 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(e\left(\frac{19}{22}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.59301 + 1.83844i −0.796507 + 0.919218i −0.998184 − 0.0602341i \(-0.980815\pi\)
0.201677 + 0.979452i \(0.435361\pi\)
\(3\) 0 0
\(4\) −0.272894 − 1.89802i −0.0682236 − 0.474505i
\(5\) 2.69128 − 1.22907i 0.538256 − 0.245813i −0.127689 − 0.991814i \(-0.540756\pi\)
0.665945 + 0.746001i \(0.268029\pi\)
\(6\) 0 0
\(7\) −1.33225 − 4.53722i −0.190321 − 0.648175i −0.998263 − 0.0589124i \(-0.981237\pi\)
0.807942 − 0.589262i \(-0.200581\pi\)
\(8\) −4.26162 − 2.73877i −0.532702 − 0.342347i
\(9\) 0 0
\(10\) −2.02769 + 6.90566i −0.202769 + 0.690566i
\(11\) −11.0247 + 9.55293i −1.00224 + 0.868448i −0.991317 − 0.131492i \(-0.958023\pi\)
−0.0109252 + 0.999940i \(0.503478\pi\)
\(12\) 0 0
\(13\) −22.7531 − 6.68090i −1.75024 − 0.513915i −0.759593 − 0.650399i \(-0.774602\pi\)
−0.990642 + 0.136484i \(0.956420\pi\)
\(14\) 10.4637 + 4.77860i 0.747406 + 0.341329i
\(15\) 0 0
\(16\) 19.1833 − 5.63274i 1.19896 − 0.352046i
\(17\) −12.1869 − 1.75221i −0.716877 − 0.103071i −0.225783 − 0.974178i \(-0.572494\pi\)
−0.491094 + 0.871106i \(0.663403\pi\)
\(18\) 0 0
\(19\) 5.54565 − 0.797344i 0.291876 − 0.0419655i 0.00517865 − 0.999987i \(-0.498352\pi\)
0.286698 + 0.958021i \(0.407442\pi\)
\(20\) −3.06723 − 4.77270i −0.153361 − 0.238635i
\(21\) 0 0
\(22\) − 35.4861i − 1.61300i
\(23\) −5.15862 − 22.4140i −0.224288 − 0.974523i
\(24\) 0 0
\(25\) −10.6391 + 12.2782i −0.425566 + 0.491129i
\(26\) 48.5283 − 31.1873i 1.86647 − 1.19951i
\(27\) 0 0
\(28\) −8.24818 + 3.76682i −0.294578 + 0.134529i
\(29\) 3.08261 − 21.4400i 0.106297 − 0.739311i −0.865057 − 0.501674i \(-0.832718\pi\)
0.971354 − 0.237637i \(-0.0763730\pi\)
\(30\) 0 0
\(31\) −0.0511435 − 0.0328679i −0.00164979 − 0.00106025i 0.539816 − 0.841783i \(-0.318494\pi\)
−0.541465 + 0.840723i \(0.682130\pi\)
\(32\) −11.7863 + 25.8083i −0.368321 + 0.806510i
\(33\) 0 0
\(34\) 22.6352 − 19.6135i 0.665742 − 0.576869i
\(35\) −9.16200 − 10.5735i −0.261771 − 0.302100i
\(36\) 0 0
\(37\) 13.3445 + 6.09421i 0.360661 + 0.164708i 0.587500 − 0.809224i \(-0.300112\pi\)
−0.226840 + 0.973932i \(0.572839\pi\)
\(38\) −7.36843 + 11.4655i −0.193906 + 0.301724i
\(39\) 0 0
\(40\) −14.8353 − 2.13300i −0.370883 − 0.0533250i
\(41\) −4.95227 − 10.8440i −0.120787 − 0.264487i 0.839574 − 0.543245i \(-0.182804\pi\)
−0.960361 + 0.278758i \(0.910077\pi\)
\(42\) 0 0
\(43\) −27.4308 − 42.6832i −0.637927 − 0.992633i −0.998211 − 0.0597907i \(-0.980957\pi\)
0.360284 − 0.932843i \(-0.382680\pi\)
\(44\) 21.1402 + 18.3181i 0.480460 + 0.416321i
\(45\) 0 0
\(46\) 49.4245 + 26.2221i 1.07445 + 0.570045i
\(47\) 23.0534 0.490498 0.245249 − 0.969460i \(-0.421130\pi\)
0.245249 + 0.969460i \(0.421130\pi\)
\(48\) 0 0
\(49\) 22.4099 − 14.4020i 0.457345 − 0.293918i
\(50\) −5.62443 − 39.1188i −0.112489 − 0.782375i
\(51\) 0 0
\(52\) −6.47131 + 45.0090i −0.124448 + 0.865557i
\(53\) 28.2602 + 96.2454i 0.533211 + 1.81595i 0.576779 + 0.816900i \(0.304309\pi\)
−0.0435676 + 0.999050i \(0.513872\pi\)
\(54\) 0 0
\(55\) −17.9293 + 39.2596i −0.325987 + 0.713811i
\(56\) −6.74889 + 22.9846i −0.120516 + 0.410440i
\(57\) 0 0
\(58\) 34.5055 + 39.8214i 0.594922 + 0.686576i
\(59\) −34.0311 − 9.99245i −0.576799 − 0.169363i −0.0196948 − 0.999806i \(-0.506269\pi\)
−0.557104 + 0.830443i \(0.688088\pi\)
\(60\) 0 0
\(61\) −49.9475 + 77.7198i −0.818811 + 1.27409i 0.140029 + 0.990147i \(0.455280\pi\)
−0.958840 + 0.283947i \(0.908356\pi\)
\(62\) 0.141898 − 0.0416649i 0.00228867 − 0.000672015i
\(63\) 0 0
\(64\) 4.55063 + 9.96449i 0.0711036 + 0.155695i
\(65\) −69.4461 + 9.98484i −1.06840 + 0.153613i
\(66\) 0 0
\(67\) −72.5767 − 62.8880i −1.08323 − 0.938628i −0.0849036 − 0.996389i \(-0.527058\pi\)
−0.998330 + 0.0577616i \(0.981604\pi\)
\(68\) 23.6092i 0.347194i
\(69\) 0 0
\(70\) 34.0339 0.486199
\(71\) −35.6457 + 41.1374i −0.502053 + 0.579400i −0.949046 − 0.315138i \(-0.897949\pi\)
0.446993 + 0.894537i \(0.352495\pi\)
\(72\) 0 0
\(73\) 6.81309 + 47.3861i 0.0933300 + 0.649124i 0.981762 + 0.190116i \(0.0608863\pi\)
−0.888432 + 0.459009i \(0.848205\pi\)
\(74\) −32.4617 + 14.8248i −0.438672 + 0.200335i
\(75\) 0 0
\(76\) −3.02675 − 10.3082i −0.0398257 − 0.135634i
\(77\) 58.0313 + 37.2945i 0.753654 + 0.484344i
\(78\) 0 0
\(79\) −10.2622 + 34.9499i −0.129901 + 0.442404i −0.998598 − 0.0529417i \(-0.983140\pi\)
0.868696 + 0.495345i \(0.164958\pi\)
\(80\) 44.7047 − 38.7368i 0.558809 − 0.484211i
\(81\) 0 0
\(82\) 27.8249 + 8.17014i 0.339329 + 0.0996359i
\(83\) −95.0493 − 43.4076i −1.14517 − 0.522983i −0.249798 − 0.968298i \(-0.580364\pi\)
−0.895374 + 0.445315i \(0.853092\pi\)
\(84\) 0 0
\(85\) −34.9519 + 10.2628i −0.411199 + 0.120739i
\(86\) 122.168 + 17.5651i 1.42056 + 0.204246i
\(87\) 0 0
\(88\) 73.1462 − 10.5168i 0.831207 − 0.119509i
\(89\) 2.45164 + 3.81483i 0.0275466 + 0.0428633i 0.854754 − 0.519034i \(-0.173708\pi\)
−0.827207 + 0.561897i \(0.810072\pi\)
\(90\) 0 0
\(91\) 112.136i 1.23227i
\(92\) −41.1346 + 15.9078i −0.447115 + 0.172911i
\(93\) 0 0
\(94\) −36.7244 + 42.3822i −0.390685 + 0.450874i
\(95\) 13.9449 − 8.96184i 0.146788 − 0.0943352i
\(96\) 0 0
\(97\) 119.559 − 54.6009i 1.23257 − 0.562896i 0.310741 − 0.950495i \(-0.399423\pi\)
0.921828 + 0.387599i \(0.126695\pi\)
\(98\) −9.22220 + 64.1418i −0.0941040 + 0.654508i
\(99\) 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 207.3.j.a.145.1 30
3.2 odd 2 23.3.d.a.7.3 ✓ 30
12.11 even 2 368.3.p.a.145.2 30
23.10 odd 22 inner 207.3.j.a.10.1 30
69.17 even 22 529.3.b.b.528.5 30
69.29 odd 22 529.3.b.b.528.6 30
69.56 even 22 23.3.d.a.10.3 yes 30
276.263 odd 22 368.3.p.a.33.2 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.3.d.a.7.3 ✓ 30 3.2 odd 2
23.3.d.a.10.3 yes 30 69.56 even 22
207.3.j.a.10.1 30 23.10 odd 22 inner
207.3.j.a.145.1 30 1.1 even 1 trivial
368.3.p.a.33.2 30 276.263 odd 22
368.3.p.a.145.2 30 12.11 even 2
529.3.b.b.528.5 30 69.17 even 22
529.3.b.b.528.6 30 69.29 odd 22