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 118.1
Root \(0.480233 + 1.05156i\) of defining polynomial
Character \(\chi\) \(=\) 207.118
Dual form 207.2.i.d.100.1

$q$-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 \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(e\left(\frac{8}{11}\right)\) \(1\)

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 −0.715372 − 0.698744i \(-0.753743\pi\)
0.489535 − 0.871984i \(-0.337167\pi\)
\(3\) 0 0
\(4\) −2.91338 + 0.855446i −1.45669 + 0.427723i
\(5\) 1.82263 − 2.10342i 0.815104 − 0.940680i −0.184004 − 0.982926i \(-0.558906\pi\)
0.999107 + 0.0422457i \(0.0134512\pi\)
\(6\) 0 0
\(7\) 2.85304 − 1.83354i 1.07835 − 0.693013i 0.124173 − 0.992261i \(-0.460372\pi\)
0.954175 + 0.299248i \(0.0967358\pi\)
\(8\) 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 0.516575 + 0.856242i \(0.327207\pi\)
−0.736881 + 0.676022i \(0.763702\pi\)
\(12\) 0 0
\(13\) −1.83466 − 1.17906i −0.508842 − 0.327013i 0.260902 − 0.965365i \(-0.415980\pi\)
−0.769744 + 0.638352i \(0.779616\pi\)
\(14\) −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 −0.0336377 − 0.999434i \(-0.510709\pi\)
−0.568633 + 0.822592i \(0.692527\pi\)
\(18\) 0 0
\(19\) −2.21869 + 0.651466i −0.509003 + 0.149457i −0.526142 − 0.850396i \(-0.676362\pi\)
0.0171398 + 0.999853i \(0.494544\pi\)
\(20\) −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 0.720603 − 0.693348i \(-0.243865\pi\)
0.231358 + 0.972869i \(0.425683\pi\)
\(30\) 0 0
\(31\) 3.71752 + 8.14024i 0.667687 + 1.46203i 0.875182 + 0.483794i \(0.160741\pi\)
−0.207496 + 0.978236i \(0.566531\pi\)
\(32\) 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 0.625221 − 0.780448i \(-0.285009\pi\)
−0.861482 + 0.507787i \(0.830464\pi\)
\(38\) 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 −0.739918 − 0.672697i \(-0.765136\pi\)
0.771151 + 0.636652i \(0.219681\pi\)
\(42\) 0 0
\(43\) 3.81341 − 8.35019i 0.581539 − 1.27339i −0.358883 − 0.933383i \(-0.616842\pi\)
0.940422 − 0.340010i \(-0.110430\pi\)
\(44\) −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 −0.920410 − 0.390954i \(-0.872145\pi\)
−0.0267281 + 0.999643i \(0.508509\pi\)
\(54\) 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 0.251803 − 0.967779i \(-0.418977\pi\)
−0.775720 + 0.631077i \(0.782613\pi\)
\(60\) 0 0
\(61\) 2.43186 + 5.32504i 0.311368 + 0.681801i 0.999021 − 0.0442383i \(-0.0140861\pi\)
−0.687653 + 0.726040i \(0.741359\pi\)
\(62\) 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 0.996231 − 0.0867422i \(-0.0276456\pi\)
−0.980315 + 0.197442i \(0.936737\pi\)
\(68\) 7.85832 0.952962
\(69\) 0 0
\(70\) −21.1831 −2.53186
\(71\) −0.189238 − 1.31618i −0.0224585 − 0.156202i 0.975506 − 0.219972i \(-0.0705967\pi\)
−0.997965 + 0.0637701i \(0.979688\pi\)
\(72\) 0 0
\(73\) −9.77086 + 2.86898i −1.14359 + 0.335789i −0.798037 − 0.602609i \(-0.794128\pi\)
−0.345556 + 0.938398i \(0.612310\pi\)
\(74\) −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 0.997632 − 0.0687729i \(-0.0219084\pi\)
0.351873 + 0.936048i \(0.385545\pi\)
\(80\) −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 0.844047 − 0.536270i \(-0.180167\pi\)
−0.650932 + 0.759136i \(0.725622\pi\)
\(84\) 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 −0.891679 − 0.452668i \(-0.850472\pi\)
0.926029 + 0.377452i \(0.123200\pi\)
\(90\) 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 −0.618809 − 0.785541i \(-0.712385\pi\)
0.865611 + 0.500717i \(0.166930\pi\)
\(98\) −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.118.1 20
3.2 odd 2 69.2.e.c.49.2 yes 20
23.8 even 11 inner 207.2.i.d.100.1 20
23.10 odd 22 4761.2.a.bu.1.9 10
23.13 even 11 4761.2.a.bt.1.9 10
69.8 odd 22 69.2.e.c.31.2 ✓ 20
69.56 even 22 1587.2.a.t.1.2 10
69.59 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 69.8 odd 22
69.2.e.c.49.2 yes 20 3.2 odd 2
207.2.i.d.100.1 20 23.8 even 11 inner
207.2.i.d.118.1 20 1.1 even 1 trivial
1587.2.a.t.1.2 10 69.56 even 22
1587.2.a.u.1.2 10 69.59 odd 22
4761.2.a.bt.1.9 10 23.13 even 11
4761.2.a.bu.1.9 10 23.10 odd 22