Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [441,6,Mod(1,441)] 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("441.1"); S:= CuspForms(chi, 6); 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, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,3,0,65,-33,0,0,375,0,-921,1137,0,-925] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{249}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.38987\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-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)
 
\(f(q)\) \(=\) \(q+9.38987 q^{2} +56.1696 q^{4} -71.7291 q^{5} +226.949 q^{8} -673.526 q^{10} +560.610 q^{11} -533.509 q^{13} +333.597 q^{16} +1005.70 q^{17} -1368.53 q^{19} -4028.99 q^{20} +5264.05 q^{22} -3228.08 q^{23} +2020.06 q^{25} -5009.58 q^{26} +753.456 q^{29} -8206.42 q^{31} -4129.95 q^{32} +9443.39 q^{34} -2808.66 q^{37} -12850.3 q^{38} -16278.9 q^{40} +245.827 q^{41} -17504.5 q^{43} +31489.2 q^{44} -30311.2 q^{46} -16345.5 q^{47} +18968.1 q^{50} -29967.0 q^{52} +29641.7 q^{53} -40212.0 q^{55} +7074.85 q^{58} -10356.1 q^{59} -954.179 q^{61} -77057.2 q^{62} -49454.8 q^{64} +38268.1 q^{65} -19815.2 q^{67} +56489.8 q^{68} -62125.4 q^{71} -27109.6 q^{73} -26373.0 q^{74} -76869.6 q^{76} +44687.4 q^{79} -23928.6 q^{80} +2308.29 q^{82} +15606.6 q^{83} -72137.9 q^{85} -164365. q^{86} +127230. q^{88} +13635.3 q^{89} -181320. q^{92} -153482. q^{94} +98163.1 q^{95} +12919.5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8} - 921 q^{10} + 1137 q^{11} - 925 q^{13} - 895 q^{16} - 324 q^{17} - 2311 q^{19} - 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} - 2508 q^{26} + 2217 q^{29}+ \cdots + 9791 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.38987 1.65991 0.829955 − 0.557831i \(-0.188366\pi\)
0.829955 + 0.557831i \(0.188366\pi\)
\(3\) 0 0
\(4\) 56.1696 1.75530
\(5\) −71.7291 −1.28313 −0.641564 − 0.767069i \(-0.721714\pi\)
−0.641564 + 0.767069i \(0.721714\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 226.949 1.25373
\(9\) 0 0
\(10\) −673.526 −2.12988
\(11\) 560.610 1.39694 0.698472 − 0.715637i \(-0.253863\pi\)
0.698472 + 0.715637i \(0.253863\pi\)
\(12\) 0 0
\(13\) −533.509 −0.875555 −0.437777 − 0.899083i \(-0.644234\pi\)
−0.437777 + 0.899083i \(0.644234\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 333.597 0.325778
\(17\) 1005.70 0.844007 0.422004 − 0.906594i \(-0.361327\pi\)
0.422004 + 0.906594i \(0.361327\pi\)
\(18\) 0 0
\(19\) −1368.53 −0.869699 −0.434850 − 0.900503i \(-0.643199\pi\)
−0.434850 + 0.900503i \(0.643199\pi\)
\(20\) −4028.99 −2.25228
\(21\) 0 0
\(22\) 5264.05 2.31880
\(23\) −3228.08 −1.27240 −0.636201 − 0.771523i \(-0.719495\pi\)
−0.636201 + 0.771523i \(0.719495\pi\)
\(24\) 0 0
\(25\) 2020.06 0.646419
\(26\) −5009.58 −1.45334
\(27\) 0 0
\(28\) 0 0
\(29\) 753.456 0.166365 0.0831827 − 0.996534i \(-0.473492\pi\)
0.0831827 + 0.996534i \(0.473492\pi\)
\(30\) 0 0
\(31\) −8206.42 −1.53373 −0.766866 − 0.641807i \(-0.778185\pi\)
−0.766866 + 0.641807i \(0.778185\pi\)
\(32\) −4129.95 −0.712968
\(33\) 0 0
\(34\) 9443.39 1.40098
\(35\) 0 0
\(36\) 0 0
\(37\) −2808.66 −0.337284 −0.168642 − 0.985677i \(-0.553938\pi\)
−0.168642 + 0.985677i \(0.553938\pi\)
\(38\) −12850.3 −1.44362
\(39\) 0 0
\(40\) −16278.9 −1.60870
\(41\) 245.827 0.0228387 0.0114193 − 0.999935i \(-0.496365\pi\)
0.0114193 + 0.999935i \(0.496365\pi\)
\(42\) 0 0
\(43\) −17504.5 −1.44371 −0.721853 − 0.692047i \(-0.756709\pi\)
−0.721853 + 0.692047i \(0.756709\pi\)
\(44\) 31489.2 2.45206
\(45\) 0 0
\(46\) −30311.2 −2.11207
\(47\) −16345.5 −1.07933 −0.539663 − 0.841881i \(-0.681449\pi\)
−0.539663 + 0.841881i \(0.681449\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 18968.1 1.07300
\(51\) 0 0
\(52\) −29967.0 −1.53686
\(53\) 29641.7 1.44948 0.724741 − 0.689021i \(-0.241959\pi\)
0.724741 + 0.689021i \(0.241959\pi\)
\(54\) 0 0
\(55\) −40212.0 −1.79246
\(56\) 0 0
\(57\) 0 0
\(58\) 7074.85 0.276151
\(59\) −10356.1 −0.387317 −0.193659 − 0.981069i \(-0.562035\pi\)
−0.193659 + 0.981069i \(0.562035\pi\)
\(60\) 0 0
\(61\) −954.179 −0.0328326 −0.0164163 − 0.999865i \(-0.505226\pi\)
−0.0164163 + 0.999865i \(0.505226\pi\)
\(62\) −77057.2 −2.54586
\(63\) 0 0
\(64\) −49454.8 −1.50924
\(65\) 38268.1 1.12345
\(66\) 0 0
\(67\) −19815.2 −0.539276 −0.269638 − 0.962962i \(-0.586904\pi\)
−0.269638 + 0.962962i \(0.586904\pi\)
\(68\) 56489.8 1.48149
\(69\) 0 0
\(70\) 0 0
\(71\) −62125.4 −1.46259 −0.731296 − 0.682060i \(-0.761084\pi\)
−0.731296 + 0.682060i \(0.761084\pi\)
\(72\) 0 0
\(73\) −27109.6 −0.595410 −0.297705 − 0.954658i \(-0.596221\pi\)
−0.297705 + 0.954658i \(0.596221\pi\)
\(74\) −26373.0 −0.559861
\(75\) 0 0
\(76\) −76869.6 −1.52658
\(77\) 0 0
\(78\) 0 0
\(79\) 44687.4 0.805595 0.402798 − 0.915289i \(-0.368038\pi\)
0.402798 + 0.915289i \(0.368038\pi\)
\(80\) −23928.6 −0.418015
\(81\) 0 0
\(82\) 2308.29 0.0379101
\(83\) 15606.6 0.248665 0.124332 − 0.992241i \(-0.460321\pi\)
0.124332 + 0.992241i \(0.460321\pi\)
\(84\) 0 0
\(85\) −72137.9 −1.08297
\(86\) −164365. −2.39642
\(87\) 0 0
\(88\) 127230. 1.75139
\(89\) 13635.3 0.182469 0.0912347 − 0.995829i \(-0.470919\pi\)
0.0912347 + 0.995829i \(0.470919\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −181320. −2.23345
\(93\) 0 0
\(94\) −153482. −1.79159
\(95\) 98163.1 1.11594
\(96\) 0 0
\(97\) 12919.5 0.139417 0.0697086 − 0.997567i \(-0.477793\pi\)
0.0697086 + 0.997567i \(0.477793\pi\)
\(98\) 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 441.6.a.s.1.2 2
3.2 odd 2 147.6.a.k.1.1 2
7.3 odd 6 63.6.e.c.37.1 4
7.5 odd 6 63.6.e.c.46.1 4
7.6 odd 2 441.6.a.t.1.2 2
21.2 odd 6 147.6.e.l.67.2 4
21.5 even 6 21.6.e.b.4.2 ✓ 4
21.11 odd 6 147.6.e.l.79.2 4
21.17 even 6 21.6.e.b.16.2 yes 4
21.20 even 2 147.6.a.i.1.1 2
84.47 odd 6 336.6.q.e.193.2 4
84.59 odd 6 336.6.q.e.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.2 ✓ 4 21.5 even 6
21.6.e.b.16.2 yes 4 21.17 even 6
63.6.e.c.37.1 4 7.3 odd 6
63.6.e.c.46.1 4 7.5 odd 6
147.6.a.i.1.1 2 21.20 even 2
147.6.a.k.1.1 2 3.2 odd 2
147.6.e.l.67.2 4 21.2 odd 6
147.6.e.l.79.2 4 21.11 odd 6
336.6.q.e.193.2 4 84.47 odd 6
336.6.q.e.289.2 4 84.59 odd 6
441.6.a.s.1.2 2 1.1 even 1 trivial
441.6.a.t.1.2 2 7.6 odd 2