Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [441,7,Mod(244,441)] 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("441.244"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 441.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-16,0,-120,0,0,0,928] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(101.453850876\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 244.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 441.244
Dual form 441.7.d.b.244.2

$q$-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)
 
\(f(q)\) \(=\) \(q-8.24264 q^{2} +3.94113 q^{4} -79.1732i q^{5} +495.044 q^{8} +652.596i q^{10} +1708.92 q^{11} +3129.09i q^{13} -4332.70 q^{16} +4076.05i q^{17} +5872.68i q^{19} -312.032i q^{20} -14086.0 q^{22} -13321.5 q^{23} +9356.60 q^{25} -25792.0i q^{26} -6510.23 q^{29} +11993.4i q^{31} +4030.09 q^{32} -33597.4i q^{34} -4640.90 q^{37} -48406.4i q^{38} -39194.2i q^{40} -19308.8i q^{41} +91636.4 q^{43} +6735.06 q^{44} +109804. q^{46} +64432.5i q^{47} -77123.1 q^{50} +12332.1i q^{52} -149600. q^{53} -135301. i q^{55} +53661.5 q^{58} +61031.7i q^{59} -98615.3i q^{61} -98857.1i q^{62} +244074. q^{64} +247740. q^{65} -311812. q^{67} +16064.2i q^{68} +401209. q^{71} +672407. i q^{73} +38253.3 q^{74} +23145.0i q^{76} -320152. q^{79} +343034. i q^{80} +159155. i q^{82} -832356. i q^{83} +322714. q^{85} -755326. q^{86} +845989. q^{88} -379497. i q^{89} -52501.7 q^{92} -531094. i q^{94} +464959. q^{95} -1.05514e6i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{2} - 120 q^{4} + 928 q^{8} + 3764 q^{11} - 496 q^{16} - 28088 q^{22} + 4940 q^{23} - 5000 q^{25} + 34544 q^{29} + 14016 q^{32} + 87740 q^{37} + 320216 q^{43} - 8664 q^{44} + 227272 q^{46} - 160000 q^{50}+ \cdots - 583500 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.24264 −1.03033 −0.515165 − 0.857091i \(-0.672269\pi\)
−0.515165 + 0.857091i \(0.672269\pi\)
\(3\) 0 0
\(4\) 3.94113 0.0615801
\(5\) − 79.1732i − 0.633386i −0.948528 − 0.316693i \(-0.897428\pi\)
0.948528 − 0.316693i \(-0.102572\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 495.044 0.966882
\(9\) 0 0
\(10\) 652.596i 0.652596i
\(11\) 1708.92 1.28394 0.641968 − 0.766732i \(-0.278118\pi\)
0.641968 + 0.766732i \(0.278118\pi\)
\(12\) 0 0
\(13\) 3129.09i 1.42426i 0.702049 + 0.712129i \(0.252269\pi\)
−0.702049 + 0.712129i \(0.747731\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4332.70 −1.05779
\(17\) 4076.05i 0.829646i 0.909902 + 0.414823i \(0.136156\pi\)
−0.909902 + 0.414823i \(0.863844\pi\)
\(18\) 0 0
\(19\) 5872.68i 0.856201i 0.903731 + 0.428101i \(0.140817\pi\)
−0.903731 + 0.428101i \(0.859183\pi\)
\(20\) − 312.032i − 0.0390039i
\(21\) 0 0
\(22\) −14086.0 −1.32288
\(23\) −13321.5 −1.09489 −0.547444 − 0.836842i \(-0.684399\pi\)
−0.547444 + 0.836842i \(0.684399\pi\)
\(24\) 0 0
\(25\) 9356.60 0.598823
\(26\) − 25792.0i − 1.46746i
\(27\) 0 0
\(28\) 0 0
\(29\) −6510.23 −0.266933 −0.133466 − 0.991053i \(-0.542611\pi\)
−0.133466 + 0.991053i \(0.542611\pi\)
\(30\) 0 0
\(31\) 11993.4i 0.402584i 0.979531 + 0.201292i \(0.0645140\pi\)
−0.979531 + 0.201292i \(0.935486\pi\)
\(32\) 4030.09 0.122989
\(33\) 0 0
\(34\) − 33597.4i − 0.854809i
\(35\) 0 0
\(36\) 0 0
\(37\) −4640.90 −0.0916214 −0.0458107 − 0.998950i \(-0.514587\pi\)
−0.0458107 + 0.998950i \(0.514587\pi\)
\(38\) − 48406.4i − 0.882170i
\(39\) 0 0
\(40\) − 39194.2i − 0.612409i
\(41\) − 19308.8i − 0.280158i −0.990140 − 0.140079i \(-0.955264\pi\)
0.990140 − 0.140079i \(-0.0447357\pi\)
\(42\) 0 0
\(43\) 91636.4 1.15256 0.576279 − 0.817253i \(-0.304504\pi\)
0.576279 + 0.817253i \(0.304504\pi\)
\(44\) 6735.06 0.0790648
\(45\) 0 0
\(46\) 109804. 1.12810
\(47\) 64432.5i 0.620599i 0.950639 + 0.310300i \(0.100429\pi\)
−0.950639 + 0.310300i \(0.899571\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −77123.1 −0.616985
\(51\) 0 0
\(52\) 12332.1i 0.0877059i
\(53\) −149600. −1.00486 −0.502428 − 0.864619i \(-0.667560\pi\)
−0.502428 + 0.864619i \(0.667560\pi\)
\(54\) 0 0
\(55\) − 135301.i − 0.813226i
\(56\) 0 0
\(57\) 0 0
\(58\) 53661.5 0.275029
\(59\) 61031.7i 0.297166i 0.988900 + 0.148583i \(0.0474713\pi\)
−0.988900 + 0.148583i \(0.952529\pi\)
\(60\) 0 0
\(61\) − 98615.3i − 0.434465i −0.976120 − 0.217233i \(-0.930297\pi\)
0.976120 − 0.217233i \(-0.0697030\pi\)
\(62\) − 98857.1i − 0.414794i
\(63\) 0 0
\(64\) 244074. 0.931069
\(65\) 247740. 0.902104
\(66\) 0 0
\(67\) −311812. −1.03674 −0.518369 − 0.855157i \(-0.673461\pi\)
−0.518369 + 0.855157i \(0.673461\pi\)
\(68\) 16064.2i 0.0510897i
\(69\) 0 0
\(70\) 0 0
\(71\) 401209. 1.12097 0.560487 − 0.828163i \(-0.310614\pi\)
0.560487 + 0.828163i \(0.310614\pi\)
\(72\) 0 0
\(73\) 672407.i 1.72848i 0.503081 + 0.864239i \(0.332200\pi\)
−0.503081 + 0.864239i \(0.667800\pi\)
\(74\) 38253.3 0.0944003
\(75\) 0 0
\(76\) 23145.0i 0.0527249i
\(77\) 0 0
\(78\) 0 0
\(79\) −320152. −0.649344 −0.324672 − 0.945827i \(-0.605254\pi\)
−0.324672 + 0.945827i \(0.605254\pi\)
\(80\) 343034.i 0.669988i
\(81\) 0 0
\(82\) 159155.i 0.288655i
\(83\) − 832356.i − 1.45571i −0.685731 − 0.727855i \(-0.740517\pi\)
0.685731 − 0.727855i \(-0.259483\pi\)
\(84\) 0 0
\(85\) 322714. 0.525486
\(86\) −755326. −1.18751
\(87\) 0 0
\(88\) 845989. 1.24141
\(89\) − 379497.i − 0.538317i −0.963096 − 0.269158i \(-0.913254\pi\)
0.963096 − 0.269158i \(-0.0867455\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −52501.7 −0.0674233
\(93\) 0 0
\(94\) − 531094.i − 0.639422i
\(95\) 464959. 0.542306
\(96\) 0 0
\(97\) − 1.05514e6i − 1.15610i −0.816001 − 0.578050i \(-0.803814\pi\)
0.816001 − 0.578050i \(-0.196186\pi\)
\(98\) 0 0
\(99\) 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 441.7.d.b.244.1 4
3.2 odd 2 49.7.b.b.48.4 4
7.2 even 3 63.7.m.b.10.2 4
7.3 odd 6 63.7.m.b.19.2 4
7.6 odd 2 inner 441.7.d.b.244.2 4
21.2 odd 6 7.7.d.b.3.1 ✓ 4
21.5 even 6 49.7.d.c.31.1 4
21.11 odd 6 49.7.d.c.19.1 4
21.17 even 6 7.7.d.b.5.1 yes 4
21.20 even 2 49.7.b.b.48.3 4
84.23 even 6 112.7.s.b.17.2 4
84.59 odd 6 112.7.s.b.33.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.b.3.1 ✓ 4 21.2 odd 6
7.7.d.b.5.1 yes 4 21.17 even 6
49.7.b.b.48.3 4 21.20 even 2
49.7.b.b.48.4 4 3.2 odd 2
49.7.d.c.19.1 4 21.11 odd 6
49.7.d.c.31.1 4 21.5 even 6
63.7.m.b.10.2 4 7.2 even 3
63.7.m.b.19.2 4 7.3 odd 6
112.7.s.b.17.2 4 84.23 even 6
112.7.s.b.33.2 4 84.59 odd 6
441.7.d.b.244.1 4 1.1 even 1 trivial
441.7.d.b.244.2 4 7.6 odd 2 inner