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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,5,Mod(449,504)] 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("504.449"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 504.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,688] 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: no
Analytic conductor: \(52.0984613943\)
Analytic rank: \(0\)
Dimension: \(12\)
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: \( x^{12} + 152x^{10} + 8601x^{8} + 233552x^{6} + 3184240x^{4} + 20126976x^{2} + 43243776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{6}\cdot 7^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.7
Root \(4.34992i\) of defining polynomial
Character \(\chi\) \(=\) 504.449
Dual form 504.5.d.b.449.6

$q$-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)
 
\(f(q)\) \(=\) \(q+1.33897i q^{5} +18.5203 q^{7} +36.5874i q^{11} -210.151 q^{13} -234.265i q^{17} +408.025 q^{19} +3.99516i q^{23} +623.207 q^{25} -508.193i q^{29} -209.310 q^{31} +24.7980i q^{35} -51.4600 q^{37} +2717.04i q^{41} -760.146 q^{43} -1972.43i q^{47} +343.000 q^{49} -4584.71i q^{53} -48.9892 q^{55} +32.3668i q^{59} +5022.10 q^{61} -281.386i q^{65} +7232.09 q^{67} -2107.67i q^{71} +2446.11 q^{73} +677.607i q^{77} +4692.65 q^{79} -8632.05i q^{83} +313.673 q^{85} +5.72611i q^{89} -3892.06 q^{91} +546.332i q^{95} +12526.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 688 q^{13} + 96 q^{19} - 1500 q^{25} - 608 q^{31} - 2728 q^{37} - 3088 q^{43} + 4116 q^{49} + 8608 q^{55} + 904 q^{61} - 15376 q^{67} + 12608 q^{73} + 11456 q^{79} + 18792 q^{85} + 3920 q^{91} - 23584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.33897i 0.0535586i 0.999641 + 0.0267793i \(0.00852514\pi\)
−0.999641 + 0.0267793i \(0.991475\pi\)
\(6\) 0 0
\(7\) 18.5203 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 36.5874i 0.302375i 0.988505 + 0.151187i \(0.0483097\pi\)
−0.988505 + 0.151187i \(0.951690\pi\)
\(12\) 0 0
\(13\) −210.151 −1.24350 −0.621750 − 0.783216i \(-0.713578\pi\)
−0.621750 + 0.783216i \(0.713578\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 234.265i − 0.810605i −0.914183 − 0.405302i \(-0.867166\pi\)
0.914183 − 0.405302i \(-0.132834\pi\)
\(18\) 0 0
\(19\) 408.025 1.13026 0.565132 − 0.825001i \(-0.308825\pi\)
0.565132 + 0.825001i \(0.308825\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.99516i 0.00755229i 0.999993 + 0.00377615i \(0.00120199\pi\)
−0.999993 + 0.00377615i \(0.998798\pi\)
\(24\) 0 0
\(25\) 623.207 0.997131
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 508.193i − 0.604272i −0.953265 − 0.302136i \(-0.902300\pi\)
0.953265 − 0.302136i \(-0.0976997\pi\)
\(30\) 0 0
\(31\) −209.310 −0.217804 −0.108902 − 0.994052i \(-0.534733\pi\)
−0.108902 + 0.994052i \(0.534733\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 24.7980i 0.0202433i
\(36\) 0 0
\(37\) −51.4600 −0.0375894 −0.0187947 − 0.999823i \(-0.505983\pi\)
−0.0187947 + 0.999823i \(0.505983\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2717.04i 1.61632i 0.588962 + 0.808161i \(0.299537\pi\)
−0.588962 + 0.808161i \(0.700463\pi\)
\(42\) 0 0
\(43\) −760.146 −0.411112 −0.205556 − 0.978645i \(-0.565900\pi\)
−0.205556 + 0.978645i \(0.565900\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1972.43i − 0.892907i −0.894807 − 0.446454i \(-0.852687\pi\)
0.894807 − 0.446454i \(-0.147313\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 4584.71i − 1.63215i −0.577947 − 0.816074i \(-0.696146\pi\)
0.577947 − 0.816074i \(-0.303854\pi\)
\(54\) 0 0
\(55\) −48.9892 −0.0161948
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 32.3668i 0.00929814i 0.999989 + 0.00464907i \(0.00147985\pi\)
−0.999989 + 0.00464907i \(0.998520\pi\)
\(60\) 0 0
\(61\) 5022.10 1.34966 0.674832 − 0.737971i \(-0.264216\pi\)
0.674832 + 0.737971i \(0.264216\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 281.386i − 0.0666001i
\(66\) 0 0
\(67\) 7232.09 1.61107 0.805534 − 0.592549i \(-0.201878\pi\)
0.805534 + 0.592549i \(0.201878\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 2107.67i − 0.418105i −0.977904 − 0.209053i \(-0.932962\pi\)
0.977904 − 0.209053i \(-0.0670380\pi\)
\(72\) 0 0
\(73\) 2446.11 0.459018 0.229509 − 0.973307i \(-0.426288\pi\)
0.229509 + 0.973307i \(0.426288\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 677.607i 0.114287i
\(78\) 0 0
\(79\) 4692.65 0.751906 0.375953 − 0.926639i \(-0.377315\pi\)
0.375953 + 0.926639i \(0.377315\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 8632.05i − 1.25302i −0.779414 − 0.626510i \(-0.784483\pi\)
0.779414 − 0.626510i \(-0.215517\pi\)
\(84\) 0 0
\(85\) 313.673 0.0434149
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.72611i 0 0.000722902i 1.00000 0.000361451i \(0.000115053\pi\)
−1.00000 0.000361451i \(0.999885\pi\)
\(90\) 0 0
\(91\) −3892.06 −0.469999
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 546.332i 0.0605354i
\(96\) 0 0
\(97\) 12526.4 1.33132 0.665662 − 0.746253i \(-0.268149\pi\)
0.665662 + 0.746253i \(0.268149\pi\)
\(98\) 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 504.5.d.b.449.7 yes 12
3.2 odd 2 inner 504.5.d.b.449.6 ✓ 12
4.3 odd 2 1008.5.d.f.449.7 12
12.11 even 2 1008.5.d.f.449.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.5.d.b.449.6 ✓ 12 3.2 odd 2 inner
504.5.d.b.449.7 yes 12 1.1 even 1 trivial
1008.5.d.f.449.6 12 12.11 even 2
1008.5.d.f.449.7 12 4.3 odd 2