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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [440,2,Mod(81,440)] 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("440.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(440, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.y (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-1,0,3,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 15 x^{10} - 22 x^{9} + 89 x^{8} - 118 x^{7} + 205 x^{6} - 68 x^{5} + 1061 x^{4} + \cdots + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 361.3
Root \(-1.51700 - 1.10216i\) of defining polynomial
Character \(\chi\) \(=\) 440.361
Dual form 440.2.y.b.401.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.45455 - 1.78334i) q^{3} +(-0.309017 - 0.951057i) q^{5} +(-0.700550 - 0.508979i) q^{7} +(1.91748 - 5.90141i) q^{9} +(-1.21410 - 3.08642i) q^{11} +(-1.37052 + 4.21802i) q^{13} +(-2.45455 - 1.78334i) q^{15} +(-0.169969 - 0.523110i) q^{17} +(0.209505 - 0.152214i) q^{19} -2.62722 q^{21} +7.92701 q^{23} +(-0.809017 + 0.587785i) q^{25} +(-3.00497 - 9.24833i) q^{27} +(7.57410 + 5.50291i) q^{29} +(-1.64270 + 5.05570i) q^{31} +(-8.48418 - 5.41063i) q^{33} +(-0.267586 + 0.823546i) q^{35} +(-5.16751 - 3.75441i) q^{37} +(4.15814 + 12.7974i) q^{39} +(-7.07258 + 5.13853i) q^{41} +6.38622 q^{43} -6.20511 q^{45} +(-0.665989 + 0.483870i) q^{47} +(-1.93141 - 5.94426i) q^{49} +(-1.35008 - 0.980890i) q^{51} +(0.741981 - 2.28358i) q^{53} +(-2.56018 + 2.10843i) q^{55} +(0.242791 - 0.747235i) q^{57} +(10.9337 + 7.94382i) q^{59} +(-2.04186 - 6.28421i) q^{61} +(-4.34699 + 3.15827i) q^{63} +4.43509 q^{65} +7.17725 q^{67} +(19.4573 - 14.1365i) q^{69} +(0.864970 + 2.66210i) q^{71} +(2.33866 + 1.69913i) q^{73} +(-0.937555 + 2.88550i) q^{75} +(-0.720387 + 2.78014i) q^{77} +(-1.82769 + 5.62505i) q^{79} +(-8.80862 - 6.39983i) q^{81} +(-2.19347 - 6.75080i) q^{83} +(-0.444984 + 0.323300i) q^{85} +28.4046 q^{87} -16.6470 q^{89} +(3.10700 - 2.25737i) q^{91} +(4.98393 + 15.3390i) q^{93} +(-0.209505 - 0.152214i) q^{95} +(-2.42082 + 7.45053i) q^{97} +(-20.5422 + 1.24672i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 3 q^{5} - 8 q^{7} + 10 q^{9} - 4 q^{11} - 7 q^{13} + q^{15} + 7 q^{17} + 3 q^{19} + 4 q^{21} + 36 q^{23} - 3 q^{25} + 8 q^{27} + 13 q^{29} + 2 q^{31} - 19 q^{33} - 2 q^{35} - 22 q^{37}+ \cdots - 79 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{3}{5}\right)\)

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 0
\(3\) 2.45455 − 1.78334i 1.41714 − 1.02961i 0.424900 − 0.905240i \(-0.360309\pi\)
0.992236 − 0.124369i \(-0.0396907\pi\)
\(4\) 0 0
\(5\) −0.309017 − 0.951057i −0.138197 − 0.425325i
\(6\) 0 0
\(7\) −0.700550 − 0.508979i −0.264783 − 0.192376i 0.447470 − 0.894299i \(-0.352325\pi\)
−0.712253 + 0.701923i \(0.752325\pi\)
\(8\) 0 0
\(9\) 1.91748 − 5.90141i 0.639161 − 1.96714i
\(10\) 0 0
\(11\) −1.21410 − 3.08642i −0.366064 − 0.930590i
\(12\) 0 0
\(13\) −1.37052 + 4.21802i −0.380114 + 1.16987i 0.559850 + 0.828594i \(0.310859\pi\)
−0.939963 + 0.341275i \(0.889141\pi\)
\(14\) 0 0
\(15\) −2.45455 − 1.78334i −0.633762 − 0.460455i
\(16\) 0 0
\(17\) −0.169969 − 0.523110i −0.0412235 − 0.126873i 0.928327 − 0.371765i \(-0.121247\pi\)
−0.969550 + 0.244892i \(0.921247\pi\)
\(18\) 0 0
\(19\) 0.209505 − 0.152214i 0.0480637 − 0.0349203i −0.563494 − 0.826120i \(-0.690543\pi\)
0.611558 + 0.791200i \(0.290543\pi\)
\(20\) 0 0
\(21\) −2.62722 −0.573306
\(22\) 0 0
\(23\) 7.92701 1.65290 0.826448 − 0.563013i \(-0.190358\pi\)
0.826448 + 0.563013i \(0.190358\pi\)
\(24\) 0 0
\(25\) −0.809017 + 0.587785i −0.161803 + 0.117557i
\(26\) 0 0
\(27\) −3.00497 − 9.24833i −0.578306 − 1.77984i
\(28\) 0 0
\(29\) 7.57410 + 5.50291i 1.40648 + 1.02186i 0.993823 + 0.110976i \(0.0353975\pi\)
0.412652 + 0.910889i \(0.364602\pi\)
\(30\) 0 0
\(31\) −1.64270 + 5.05570i −0.295037 + 0.908031i 0.688172 + 0.725548i \(0.258413\pi\)
−0.983209 + 0.182483i \(0.941587\pi\)
\(32\) 0 0
\(33\) −8.48418 − 5.41063i −1.47691 − 0.941869i
\(34\) 0 0
\(35\) −0.267586 + 0.823546i −0.0452303 + 0.139205i
\(36\) 0 0
\(37\) −5.16751 − 3.75441i −0.849533 − 0.617222i 0.0754844 − 0.997147i \(-0.475950\pi\)
−0.925017 + 0.379925i \(0.875950\pi\)
\(38\) 0 0
\(39\) 4.15814 + 12.7974i 0.665836 + 2.04923i
\(40\) 0 0
\(41\) −7.07258 + 5.13853i −1.10455 + 0.802503i −0.981797 − 0.189934i \(-0.939173\pi\)
−0.122754 + 0.992437i \(0.539173\pi\)
\(42\) 0 0
\(43\) 6.38622 0.973890 0.486945 − 0.873433i \(-0.338111\pi\)
0.486945 + 0.873433i \(0.338111\pi\)
\(44\) 0 0
\(45\) −6.20511 −0.925003
\(46\) 0 0
\(47\) −0.665989 + 0.483870i −0.0971445 + 0.0705796i −0.635297 − 0.772268i \(-0.719122\pi\)
0.538153 + 0.842847i \(0.319122\pi\)
\(48\) 0 0
\(49\) −1.93141 − 5.94426i −0.275916 − 0.849181i
\(50\) 0 0
\(51\) −1.35008 − 0.980890i −0.189049 − 0.137352i
\(52\) 0 0
\(53\) 0.741981 − 2.28358i 0.101919 − 0.313674i −0.887076 − 0.461623i \(-0.847267\pi\)
0.988995 + 0.147949i \(0.0472672\pi\)
\(54\) 0 0
\(55\) −2.56018 + 2.10843i −0.345215 + 0.284301i
\(56\) 0 0
\(57\) 0.242791 − 0.747235i 0.0321585 − 0.0989737i
\(58\) 0 0
\(59\) 10.9337 + 7.94382i 1.42345 + 1.03420i 0.991190 + 0.132449i \(0.0422840\pi\)
0.432261 + 0.901749i \(0.357716\pi\)
\(60\) 0 0
\(61\) −2.04186 − 6.28421i −0.261434 − 0.804611i −0.992494 − 0.122297i \(-0.960974\pi\)
0.731060 − 0.682313i \(-0.239026\pi\)
\(62\) 0 0
\(63\) −4.34699 + 3.15827i −0.547669 + 0.397905i
\(64\) 0 0
\(65\) 4.43509 0.550105
\(66\) 0 0
\(67\) 7.17725 0.876840 0.438420 − 0.898770i \(-0.355538\pi\)
0.438420 + 0.898770i \(0.355538\pi\)
\(68\) 0 0
\(69\) 19.4573 − 14.1365i 2.34238 − 1.70184i
\(70\) 0 0
\(71\) 0.864970 + 2.66210i 0.102653 + 0.315934i 0.989172 − 0.146758i \(-0.0468838\pi\)
−0.886519 + 0.462691i \(0.846884\pi\)
\(72\) 0 0
\(73\) 2.33866 + 1.69913i 0.273719 + 0.198868i 0.716173 − 0.697923i \(-0.245892\pi\)
−0.442454 + 0.896791i \(0.645892\pi\)
\(74\) 0 0
\(75\) −0.937555 + 2.88550i −0.108260 + 0.333189i
\(76\) 0 0
\(77\) −0.720387 + 2.78014i −0.0820958 + 0.316826i
\(78\) 0 0
\(79\) −1.82769 + 5.62505i −0.205631 + 0.632867i 0.794056 + 0.607845i \(0.207966\pi\)
−0.999687 + 0.0250225i \(0.992034\pi\)
\(80\) 0 0
\(81\) −8.80862 − 6.39983i −0.978735 − 0.711093i
\(82\) 0 0
\(83\) −2.19347 − 6.75080i −0.240764 − 0.740996i −0.996304 − 0.0858936i \(-0.972625\pi\)
0.755540 − 0.655102i \(-0.227375\pi\)
\(84\) 0 0
\(85\) −0.444984 + 0.323300i −0.0482653 + 0.0350668i
\(86\) 0 0
\(87\) 28.4046 3.04529
\(88\) 0 0
\(89\) −16.6470 −1.76458 −0.882291 − 0.470704i \(-0.844000\pi\)
−0.882291 + 0.470704i \(0.844000\pi\)
\(90\) 0 0
\(91\) 3.10700 − 2.25737i 0.325702 − 0.236637i
\(92\) 0 0
\(93\) 4.98393 + 15.3390i 0.516809 + 1.59058i
\(94\) 0 0
\(95\) −0.209505 − 0.152214i −0.0214948 − 0.0156169i
\(96\) 0 0
\(97\) −2.42082 + 7.45053i −0.245797 + 0.756486i 0.749707 + 0.661770i \(0.230194\pi\)
−0.995504 + 0.0947164i \(0.969806\pi\)
\(98\) 0 0
\(99\) −20.5422 + 1.24672i −2.06457 + 0.125300i
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 440.2.y.b.361.3 ✓ 12
4.3 odd 2 880.2.bo.j.801.1 12
11.4 even 5 4840.2.a.bf.1.1 6
11.5 even 5 inner 440.2.y.b.401.3 yes 12
11.7 odd 10 4840.2.a.be.1.1 6
44.7 even 10 9680.2.a.cy.1.6 6
44.15 odd 10 9680.2.a.cx.1.6 6
44.27 odd 10 880.2.bo.j.401.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.361.3 ✓ 12 1.1 even 1 trivial
440.2.y.b.401.3 yes 12 11.5 even 5 inner
880.2.bo.j.401.1 12 44.27 odd 10
880.2.bo.j.801.1 12 4.3 odd 2
4840.2.a.be.1.1 6 11.7 odd 10
4840.2.a.bf.1.1 6 11.4 even 5
9680.2.a.cx.1.6 6 44.15 odd 10
9680.2.a.cy.1.6 6 44.7 even 10