Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [207,2,Mod(55,207)] 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("207.55"); S:= CuspForms(chi, 2); 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, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,4] 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: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 7 x^{19} + 24 x^{18} - 70 x^{17} + 209 x^{16} - 527 x^{15} + 1115 x^{14} - 2187 x^{13} + \cdots + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 69)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 100.1
Root \(0.480233 - 1.05156i\) of defining polynomial
Character \(\chi\) \(=\) 207.100
Dual form 207.2.i.d.118.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.319381 + 2.22134i) q^{2} +(-2.91338 - 0.855446i) q^{4} +(1.82263 + 2.10342i) q^{5} +(2.85304 + 1.83354i) q^{7} +(0.966182 - 2.11564i) q^{8} +(-5.25454 + 3.37689i) q^{10} +(-0.730673 - 5.08195i) q^{11} +(-1.83466 + 1.17906i) q^{13} +(-4.98413 + 5.75199i) q^{14} +(-0.717732 - 0.461258i) q^{16} +(-2.48322 + 0.729141i) q^{17} +(-2.21869 - 0.651466i) q^{19} +(-3.51064 - 7.68724i) q^{20} +11.5221 q^{22} +(3.18567 + 3.58490i) q^{23} +(-0.390848 + 2.71841i) q^{25} +(-2.03315 - 4.45197i) q^{26} +(-6.74351 - 7.78242i) q^{28} +(5.12647 - 1.50527i) q^{29} +(3.71752 - 8.14024i) q^{31} +(4.30002 - 4.96249i) q^{32} +(-0.826577 - 5.74897i) q^{34} +(1.34332 + 9.34301i) q^{35} +(-1.43712 + 1.65853i) q^{37} +(2.15574 - 4.72041i) q^{38} +(6.21109 - 1.82374i) q^{40} +(0.199985 + 0.230795i) q^{41} +(3.81341 + 8.35019i) q^{43} +(-2.21860 + 15.4307i) q^{44} +(-8.98074 + 5.93153i) q^{46} +4.99153 q^{47} +(1.87008 + 4.09490i) q^{49} +(-5.91369 - 1.73642i) q^{50} +(6.35368 - 1.86561i) q^{52} +(-6.89527 - 4.43132i) q^{53} +(9.35774 - 10.7994i) q^{55} +(6.63567 - 4.26449i) q^{56} +(1.70642 + 11.8684i) q^{58} +(-4.02428 + 2.58625i) q^{59} +(2.43186 - 5.32504i) q^{61} +(16.8950 + 10.8577i) q^{62} +(8.53265 + 9.84720i) q^{64} +(-5.82396 - 1.71007i) q^{65} +(0.130280 - 0.906118i) q^{67} +7.85832 q^{68} -21.1831 q^{70} +(-0.189238 + 1.31618i) q^{71} +(-9.77086 - 2.86898i) q^{73} +(-3.22518 - 3.72205i) q^{74} +(5.90660 + 3.79594i) q^{76} +(7.23330 - 15.8387i) q^{77} +(11.9947 - 7.70851i) q^{79} +(-0.337936 - 2.35040i) q^{80} +(-0.576546 + 0.370523i) q^{82} +(1.75936 - 2.03041i) q^{83} +(-6.05968 - 3.89432i) q^{85} +(-19.7666 + 5.80399i) q^{86} +(-11.4576 - 3.36424i) q^{88} +(0.324058 + 0.709587i) q^{89} -7.39621 q^{91} +(-6.21440 - 13.1693i) q^{92} +(-1.59420 + 11.0879i) q^{94} +(-2.67354 - 5.85423i) q^{95} +(2.43072 + 2.80520i) q^{97} +(-9.69346 + 2.84626i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{2} - 6 q^{4} + 6 q^{5} - 6 q^{7} - 10 q^{8} - 18 q^{10} + 16 q^{11} + 14 q^{13} + 22 q^{14} - 8 q^{16} - 11 q^{17} - 11 q^{19} - 57 q^{20} + 26 q^{22} - 4 q^{25} + 14 q^{26} - 14 q^{28} - 12 q^{29}+ \cdots - 85 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(47\)
\(\chi(n)\) \(e\left(\frac{3}{11}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.319381 + 2.22134i −0.225837 + 1.57073i 0.489535 + 0.871984i \(0.337167\pi\)
−0.715372 + 0.698744i \(0.753743\pi\)
\(3\) 0 0
\(4\) −2.91338 − 0.855446i −1.45669 − 0.427723i
\(5\) 1.82263 + 2.10342i 0.815104 + 0.940680i 0.999107 − 0.0422457i \(-0.0134512\pi\)
−0.184004 + 0.982926i \(0.558906\pi\)
\(6\) 0 0
\(7\) 2.85304 + 1.83354i 1.07835 + 0.693013i 0.954175 − 0.299248i \(-0.0967358\pi\)
0.124173 + 0.992261i \(0.460372\pi\)
\(8\) 0.966182 − 2.11564i 0.341597 − 0.747993i
\(9\) 0 0
\(10\) −5.25454 + 3.37689i −1.66163 + 1.06787i
\(11\) −0.730673 − 5.08195i −0.220306 − 1.53226i −0.736881 − 0.676022i \(-0.763702\pi\)
0.516575 − 0.856242i \(-0.327207\pi\)
\(12\) 0 0
\(13\) −1.83466 + 1.17906i −0.508842 + 0.327013i −0.769744 − 0.638352i \(-0.779616\pi\)
0.260902 + 0.965365i \(0.415980\pi\)
\(14\) −4.98413 + 5.75199i −1.33206 + 1.53728i
\(15\) 0 0
\(16\) −0.717732 − 0.461258i −0.179433 − 0.115315i
\(17\) −2.48322 + 0.729141i −0.602270 + 0.176843i −0.568633 − 0.822592i \(-0.692527\pi\)
−0.0336377 + 0.999434i \(0.510709\pi\)
\(18\) 0 0
\(19\) −2.21869 − 0.651466i −0.509003 − 0.149457i 0.0171398 − 0.999853i \(-0.494544\pi\)
−0.526142 + 0.850396i \(0.676362\pi\)
\(20\) −3.51064 − 7.68724i −0.785004 − 1.71892i
\(21\) 0 0
\(22\) 11.5221 2.45652
\(23\) 3.18567 + 3.58490i 0.664259 + 0.747503i
\(24\) 0 0
\(25\) −0.390848 + 2.71841i −0.0781696 + 0.543682i
\(26\) −2.03315 − 4.45197i −0.398733 − 0.873104i
\(27\) 0 0
\(28\) −6.74351 − 7.78242i −1.27440 − 1.47074i
\(29\) 5.12647 − 1.50527i 0.951961 − 0.279521i 0.231358 − 0.972869i \(-0.425683\pi\)
0.720603 + 0.693348i \(0.243865\pi\)
\(30\) 0 0
\(31\) 3.71752 − 8.14024i 0.667687 − 1.46203i −0.207496 − 0.978236i \(-0.566531\pi\)
0.875182 − 0.483794i \(-0.160741\pi\)
\(32\) 4.30002 − 4.96249i 0.760144 − 0.877253i
\(33\) 0 0
\(34\) −0.826577 − 5.74897i −0.141757 − 0.985941i
\(35\) 1.34332 + 9.34301i 0.227063 + 1.57926i
\(36\) 0 0
\(37\) −1.43712 + 1.65853i −0.236262 + 0.272661i −0.861482 − 0.507787i \(-0.830464\pi\)
0.625221 + 0.780448i \(0.285009\pi\)
\(38\) 2.15574 − 4.72041i 0.349707 − 0.765752i
\(39\) 0 0
\(40\) 6.21109 − 1.82374i 0.982059 − 0.288359i
\(41\) 0.199985 + 0.230795i 0.0312323 + 0.0360440i 0.771151 − 0.636652i \(-0.219681\pi\)
−0.739918 + 0.672697i \(0.765136\pi\)
\(42\) 0 0
\(43\) 3.81341 + 8.35019i 0.581539 + 1.27339i 0.940422 + 0.340010i \(0.110430\pi\)
−0.358883 + 0.933383i \(0.616842\pi\)
\(44\) −2.21860 + 15.4307i −0.334467 + 2.32627i
\(45\) 0 0
\(46\) −8.98074 + 5.93153i −1.32414 + 0.874557i
\(47\) 4.99153 0.728090 0.364045 − 0.931381i \(-0.381395\pi\)
0.364045 + 0.931381i \(0.381395\pi\)
\(48\) 0 0
\(49\) 1.87008 + 4.09490i 0.267154 + 0.584986i
\(50\) −5.91369 − 1.73642i −0.836323 − 0.245567i
\(51\) 0 0
\(52\) 6.35368 − 1.86561i 0.881097 − 0.258713i
\(53\) −6.89527 − 4.43132i −0.947138 − 0.608689i −0.0267281 − 0.999643i \(-0.508509\pi\)
−0.920410 + 0.390954i \(0.872145\pi\)
\(54\) 0 0
\(55\) 9.35774 − 10.7994i 1.26180 − 1.45619i
\(56\) 6.63567 − 4.26449i 0.886729 − 0.569866i
\(57\) 0 0
\(58\) 1.70642 + 11.8684i 0.224064 + 1.55840i
\(59\) −4.02428 + 2.58625i −0.523917 + 0.336701i −0.775720 − 0.631077i \(-0.782613\pi\)
0.251803 + 0.967779i \(0.418977\pi\)
\(60\) 0 0
\(61\) 2.43186 − 5.32504i 0.311368 − 0.681801i −0.687653 − 0.726040i \(-0.741359\pi\)
0.999021 + 0.0442383i \(0.0140861\pi\)
\(62\) 16.8950 + 10.8577i 2.14566 + 1.37893i
\(63\) 0 0
\(64\) 8.53265 + 9.84720i 1.06658 + 1.23090i
\(65\) −5.82396 − 1.71007i −0.722374 − 0.212108i
\(66\) 0 0
\(67\) 0.130280 − 0.906118i 0.0159162 − 0.110700i −0.980315 − 0.197442i \(-0.936737\pi\)
0.996231 + 0.0867422i \(0.0276456\pi\)
\(68\) 7.85832 0.952962
\(69\) 0 0
\(70\) −21.1831 −2.53186
\(71\) −0.189238 + 1.31618i −0.0224585 + 0.156202i −0.997965 − 0.0637701i \(-0.979688\pi\)
0.975506 + 0.219972i \(0.0705967\pi\)
\(72\) 0 0
\(73\) −9.77086 − 2.86898i −1.14359 − 0.335789i −0.345556 − 0.938398i \(-0.612310\pi\)
−0.798037 + 0.602609i \(0.794128\pi\)
\(74\) −3.22518 − 3.72205i −0.374919 − 0.432680i
\(75\) 0 0
\(76\) 5.90660 + 3.79594i 0.677534 + 0.435424i
\(77\) 7.23330 − 15.8387i 0.824311 − 1.80499i
\(78\) 0 0
\(79\) 11.9947 − 7.70851i 1.34951 − 0.867275i 0.351873 − 0.936048i \(-0.385545\pi\)
0.997632 + 0.0687729i \(0.0219084\pi\)
\(80\) −0.337936 − 2.35040i −0.0377824 − 0.262782i
\(81\) 0 0
\(82\) −0.576546 + 0.370523i −0.0636688 + 0.0409175i
\(83\) 1.75936 − 2.03041i 0.193115 − 0.222866i −0.650932 − 0.759136i \(-0.725622\pi\)
0.844047 + 0.536270i \(0.180167\pi\)
\(84\) 0 0
\(85\) −6.05968 − 3.89432i −0.657265 − 0.422399i
\(86\) −19.7666 + 5.80399i −2.13149 + 0.625861i
\(87\) 0 0
\(88\) −11.4576 − 3.36424i −1.22138 − 0.358629i
\(89\) 0.324058 + 0.709587i 0.0343500 + 0.0752161i 0.926029 − 0.377452i \(-0.123200\pi\)
−0.891679 + 0.452668i \(0.850472\pi\)
\(90\) 0 0
\(91\) −7.39621 −0.775333
\(92\) −6.21440 − 13.1693i −0.647896 − 1.37300i
\(93\) 0 0
\(94\) −1.59420 + 11.0879i −0.164429 + 1.14363i
\(95\) −2.67354 − 5.85423i −0.274299 − 0.600631i
\(96\) 0 0
\(97\) 2.43072 + 2.80520i 0.246802 + 0.284825i 0.865611 − 0.500717i \(-0.166930\pi\)
−0.618809 + 0.785541i \(0.712385\pi\)
\(98\) −9.69346 + 2.84626i −0.979187 + 0.287515i
\(99\) 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 207.2.i.d.100.1 20
3.2 odd 2 69.2.e.c.31.2 ✓ 20
23.3 even 11 inner 207.2.i.d.118.1 20
23.7 odd 22 4761.2.a.bu.1.9 10
23.16 even 11 4761.2.a.bt.1.9 10
69.26 odd 22 69.2.e.c.49.2 yes 20
69.53 even 22 1587.2.a.t.1.2 10
69.62 odd 22 1587.2.a.u.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.e.c.31.2 ✓ 20 3.2 odd 2
69.2.e.c.49.2 yes 20 69.26 odd 22
207.2.i.d.100.1 20 1.1 even 1 trivial
207.2.i.d.118.1 20 23.3 even 11 inner
1587.2.a.t.1.2 10 69.53 even 22
1587.2.a.u.1.2 10 69.62 odd 22
4761.2.a.bt.1.9 10 23.16 even 11
4761.2.a.bu.1.9 10 23.7 odd 22