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 127.1
Root \(0.124087 + 0.863041i\) of defining polynomial
Character \(\chi\) \(=\) 207.127
Dual form 207.2.i.d.163.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.19300 + 0.766692i) q^{2} +(0.00459227 - 0.0100557i) q^{4} +(-0.815022 + 0.239312i) q^{5} +(-3.31578 - 3.82661i) q^{7} +(-0.401407 - 2.79185i) q^{8} +(0.788839 - 0.910368i) q^{10} +(1.24362 + 0.799227i) q^{11} +(-0.666743 + 0.769463i) q^{13} +(6.88954 + 2.02295i) q^{14} +(2.63384 + 3.03962i) q^{16} +(-2.37590 - 5.20249i) q^{17} +(1.89731 - 4.15452i) q^{19} +(-0.00133636 + 0.00929457i) q^{20} -2.09640 q^{22} +(-4.33154 - 2.05857i) q^{23} +(-3.59928 + 2.31312i) q^{25} +(0.205481 - 1.42915i) q^{26} +(-0.0537061 + 0.0157695i) q^{28} +(1.19571 + 2.61825i) q^{29} +(-0.751167 - 5.22448i) q^{31} +(-0.0600008 - 0.0176178i) q^{32} +(6.82314 + 4.38497i) q^{34} +(3.61819 + 2.32527i) q^{35} +(0.443467 + 0.130214i) q^{37} +(0.921759 + 6.41098i) q^{38} +(0.995278 + 2.17935i) q^{40} +(-6.61755 + 1.94309i) q^{41} +(-0.690297 + 4.80112i) q^{43} +(0.0137478 - 0.00883519i) q^{44} +(6.74580 - 0.865089i) q^{46} -4.11922 q^{47} +(-2.65237 + 18.4477i) q^{49} +(2.52048 - 5.51907i) q^{50} +(0.00467560 + 0.0102381i) q^{52} +(1.84980 + 2.13478i) q^{53} +(-1.20484 - 0.353774i) q^{55} +(-9.35234 + 10.7932i) q^{56} +(-3.43387 - 2.20681i) q^{58} +(1.38652 - 1.60013i) q^{59} +(-1.39065 - 9.67215i) q^{61} +(4.90170 + 5.65687i) q^{62} +(-7.63305 + 2.24127i) q^{64} +(0.359269 - 0.786689i) q^{65} +(-4.61302 + 2.96461i) q^{67} -0.0632253 q^{68} -6.09924 q^{70} +(10.6456 - 6.84151i) q^{71} +(0.730756 - 1.60013i) q^{73} +(-0.628889 + 0.184658i) q^{74} +(-0.0330636 - 0.0381574i) q^{76} +(-1.06524 - 7.40892i) q^{77} +(1.53299 - 1.76917i) q^{79} +(-2.87405 - 1.84704i) q^{80} +(6.40496 - 7.39172i) q^{82} +(15.7737 + 4.63157i) q^{83} +(3.18143 + 3.67156i) q^{85} +(-2.85746 - 6.25696i) q^{86} +(1.73212 - 3.79282i) q^{88} +(-1.44953 + 10.0817i) q^{89} +5.15521 q^{91} +(-0.0405920 + 0.0341030i) q^{92} +(4.91421 - 3.15817i) q^{94} +(-0.552119 + 3.84008i) q^{95} +(10.8134 - 3.17511i) q^{97} +(-10.9794 - 24.0415i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{2} - 6 q^{4} + 6 q^{5} - 6 q^{7} - 10 q^{8} - 18 q^{10} + 16 q^{11} + 14 q^{13} + 22 q^{14} - 8 q^{16} - 11 q^{17} - 11 q^{19} - 57 q^{20} + 26 q^{22} - 4 q^{25} + 14 q^{26} - 14 q^{28} - 12 q^{29}+ \cdots - 85 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(47\)
\(\chi(n)\) \(e\left(\frac{10}{11}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.19300 + 0.766692i −0.843575 + 0.542133i −0.889565 − 0.456809i \(-0.848992\pi\)
0.0459895 + 0.998942i \(0.485356\pi\)
\(3\) 0 0
\(4\) 0.00459227 − 0.0100557i 0.00229614 − 0.00502784i
\(5\) −0.815022 + 0.239312i −0.364489 + 0.107024i −0.458848 − 0.888515i \(-0.651738\pi\)
0.0943595 + 0.995538i \(0.469920\pi\)
\(6\) 0 0
\(7\) −3.31578 − 3.82661i −1.25325 − 1.44632i −0.846155 − 0.532937i \(-0.821088\pi\)
−0.407092 − 0.913387i \(-0.633457\pi\)
\(8\) −0.401407 − 2.79185i −0.141919 − 0.987067i
\(9\) 0 0
\(10\) 0.788839 − 0.910368i 0.249453 − 0.287884i
\(11\) 1.24362 + 0.799227i 0.374966 + 0.240976i 0.714527 − 0.699608i \(-0.246642\pi\)
−0.339561 + 0.940584i \(0.610278\pi\)
\(12\) 0 0
\(13\) −0.666743 + 0.769463i −0.184921 + 0.213411i −0.840639 − 0.541595i \(-0.817821\pi\)
0.655718 + 0.755006i \(0.272366\pi\)
\(14\) 6.88954 + 2.02295i 1.84131 + 0.540657i
\(15\) 0 0
\(16\) 2.63384 + 3.03962i 0.658460 + 0.759904i
\(17\) −2.37590 − 5.20249i −0.576240 − 1.26179i −0.943408 − 0.331635i \(-0.892400\pi\)
0.367168 − 0.930155i \(-0.380327\pi\)
\(18\) 0 0
\(19\) 1.89731 − 4.15452i 0.435272 − 0.953113i −0.557170 − 0.830399i \(-0.688113\pi\)
0.992442 − 0.122715i \(-0.0391599\pi\)
\(20\) −0.00133636 + 0.00929457i −0.000298819 + 0.00207833i
\(21\) 0 0
\(22\) −2.09640 −0.446953
\(23\) −4.33154 − 2.05857i −0.903189 − 0.429242i
\(24\) 0 0
\(25\) −3.59928 + 2.31312i −0.719856 + 0.462623i
\(26\) 0.205481 − 1.42915i 0.0402982 − 0.280280i
\(27\) 0 0
\(28\) −0.0537061 + 0.0157695i −0.0101495 + 0.00298016i
\(29\) 1.19571 + 2.61825i 0.222039 + 0.486197i 0.987566 − 0.157207i \(-0.0502490\pi\)
−0.765527 + 0.643404i \(0.777522\pi\)
\(30\) 0 0
\(31\) −0.751167 − 5.22448i −0.134913 − 0.938344i −0.939021 − 0.343860i \(-0.888265\pi\)
0.804107 − 0.594484i \(-0.202644\pi\)
\(32\) −0.0600008 − 0.0176178i −0.0106067 − 0.00311442i
\(33\) 0 0
\(34\) 6.82314 + 4.38497i 1.17016 + 0.752016i
\(35\) 3.61819 + 2.32527i 0.611585 + 0.393042i
\(36\) 0 0
\(37\) 0.443467 + 0.130214i 0.0729056 + 0.0214070i 0.317982 − 0.948097i \(-0.396995\pi\)
−0.245076 + 0.969504i \(0.578813\pi\)
\(38\) 0.921759 + 6.41098i 0.149529 + 1.04000i
\(39\) 0 0
\(40\) 0.995278 + 2.17935i 0.157367 + 0.344586i
\(41\) −6.61755 + 1.94309i −1.03349 + 0.303460i −0.754128 − 0.656727i \(-0.771940\pi\)
−0.279360 + 0.960186i \(0.590122\pi\)
\(42\) 0 0
\(43\) −0.690297 + 4.80112i −0.105269 + 0.732164i 0.867001 + 0.498306i \(0.166044\pi\)
−0.972271 + 0.233859i \(0.924865\pi\)
\(44\) 0.0137478 − 0.00883519i 0.00207256 − 0.00133195i
\(45\) 0 0
\(46\) 6.74580 − 0.865089i 0.994615 − 0.127550i
\(47\) −4.11922 −0.600850 −0.300425 − 0.953805i \(-0.597129\pi\)
−0.300425 + 0.953805i \(0.597129\pi\)
\(48\) 0 0
\(49\) −2.65237 + 18.4477i −0.378911 + 2.63538i
\(50\) 2.52048 − 5.51907i 0.356449 − 0.780515i
\(51\) 0 0
\(52\) 0.00467560 + 0.0102381i 0.000648389 + 0.00141977i
\(53\) 1.84980 + 2.13478i 0.254090 + 0.293235i 0.868436 − 0.495801i \(-0.165126\pi\)
−0.614346 + 0.789037i \(0.710580\pi\)
\(54\) 0 0
\(55\) −1.20484 − 0.353774i −0.162461 − 0.0477029i
\(56\) −9.35234 + 10.7932i −1.24976 + 1.44230i
\(57\) 0 0
\(58\) −3.43387 − 2.20681i −0.450889 − 0.289769i
\(59\) 1.38652 − 1.60013i 0.180510 − 0.208320i −0.658282 − 0.752771i \(-0.728717\pi\)
0.838792 + 0.544452i \(0.183262\pi\)
\(60\) 0 0
\(61\) −1.39065 − 9.67215i −0.178054 − 1.23839i −0.861259 − 0.508166i \(-0.830324\pi\)
0.683205 − 0.730226i \(-0.260585\pi\)
\(62\) 4.90170 + 5.65687i 0.622517 + 0.718423i
\(63\) 0 0
\(64\) −7.63305 + 2.24127i −0.954131 + 0.280158i
\(65\) 0.359269 − 0.786689i 0.0445618 − 0.0975767i
\(66\) 0 0
\(67\) −4.61302 + 2.96461i −0.563571 + 0.362185i −0.791195 − 0.611564i \(-0.790541\pi\)
0.227624 + 0.973749i \(0.426904\pi\)
\(68\) −0.0632253 −0.00766719
\(69\) 0 0
\(70\) −6.09924 −0.728999
\(71\) 10.6456 − 6.84151i 1.26340 − 0.811938i 0.274654 − 0.961543i \(-0.411437\pi\)
0.988746 + 0.149606i \(0.0478004\pi\)
\(72\) 0 0
\(73\) 0.730756 − 1.60013i 0.0855285 − 0.187281i −0.862035 − 0.506849i \(-0.830810\pi\)
0.947564 + 0.319567i \(0.103538\pi\)
\(74\) −0.628889 + 0.184658i −0.0731068 + 0.0214661i
\(75\) 0 0
\(76\) −0.0330636 − 0.0381574i −0.00379265 − 0.00437695i
\(77\) −1.06524 − 7.40892i −0.121396 − 0.844325i
\(78\) 0 0
\(79\) 1.53299 − 1.76917i 0.172475 − 0.199047i −0.662930 − 0.748681i \(-0.730687\pi\)
0.835405 + 0.549634i \(0.185233\pi\)
\(80\) −2.87405 − 1.84704i −0.321329 − 0.206506i
\(81\) 0 0
\(82\) 6.40496 − 7.39172i 0.707310 − 0.816279i
\(83\) 15.7737 + 4.63157i 1.73139 + 0.508381i 0.987186 − 0.159576i \(-0.0510128\pi\)
0.744200 + 0.667957i \(0.232831\pi\)
\(84\) 0 0
\(85\) 3.18143 + 3.67156i 0.345074 + 0.398237i
\(86\) −2.85746 − 6.25696i −0.308128 − 0.674706i
\(87\) 0 0
\(88\) 1.73212 − 3.79282i 0.184645 − 0.404316i
\(89\) −1.44953 + 10.0817i −0.153650 + 1.06866i 0.756386 + 0.654126i \(0.226963\pi\)
−0.910035 + 0.414531i \(0.863946\pi\)
\(90\) 0 0
\(91\) 5.15521 0.540413
\(92\) −0.0405920 + 0.0341030i −0.00423200 + 0.00355549i
\(93\) 0 0
\(94\) 4.91421 − 3.15817i 0.506862 − 0.325741i
\(95\) −0.552119 + 3.84008i −0.0566463 + 0.393983i
\(96\) 0 0
\(97\) 10.8134 − 3.17511i 1.09794 − 0.322384i 0.317905 − 0.948122i \(-0.397021\pi\)
0.780033 + 0.625739i \(0.215202\pi\)
\(98\) −10.9794 − 24.0415i −1.10909 − 2.42856i
\(99\) 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.127.1 20
3.2 odd 2 69.2.e.c.58.2 yes 20
23.2 even 11 inner 207.2.i.d.163.1 20
23.5 odd 22 4761.2.a.bu.1.3 10
23.18 even 11 4761.2.a.bt.1.3 10
69.2 odd 22 69.2.e.c.25.2 ✓ 20
69.5 even 22 1587.2.a.t.1.8 10
69.41 odd 22 1587.2.a.u.1.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.e.c.25.2 ✓ 20 69.2 odd 22
69.2.e.c.58.2 yes 20 3.2 odd 2
207.2.i.d.127.1 20 1.1 even 1 trivial
207.2.i.d.163.1 20 23.2 even 11 inner
1587.2.a.t.1.8 10 69.5 even 22
1587.2.a.u.1.8 10 69.41 odd 22
4761.2.a.bt.1.3 10 23.18 even 11
4761.2.a.bu.1.3 10 23.5 odd 22