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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3328,2,Mod(1665,3328)] 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("3328.1665"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,-10,0,-16,0,0,0,0,0,18,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.5742137927\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.89857052655616.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 16x^{8} + 88x^{6} + 185x^{4} + 100x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1664)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1665.3
Root \(-0.857815i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1665
Dual form 3328.2.b.bc.1665.8

$q$-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)
 
\(f(q)\) \(=\) \(q-1.40634i q^{3} +0.422133i q^{5} +4.72205 q^{7} +1.02221 q^{9} +3.71563i q^{11} +1.00000i q^{13} +0.593662 q^{15} +2.39054 q^{17} -8.15998i q^{19} -6.64080i q^{21} -7.87561 q^{23} +4.82180 q^{25} -5.65659i q^{27} -5.87561i q^{29} -0.528307 q^{31} +5.22543 q^{33} +1.99333i q^{35} -1.20954i q^{37} +1.40634 q^{39} -9.03134 q^{41} +2.19374i q^{43} +0.431510i q^{45} +11.1348 q^{47} +15.2977 q^{49} -3.36191i q^{51} +5.03134i q^{53} -1.56849 q^{55} -11.4757 q^{57} +3.71563i q^{59} +14.6567i q^{61} +4.82694 q^{63} -0.422133 q^{65} -6.47144i q^{67} +11.0758i q^{69} +9.34063 q^{71} +16.5070 q^{73} -6.78109i q^{75} +17.5454i q^{77} +11.4313 q^{79} -4.88844 q^{81} -6.08396i q^{83} +1.00913i q^{85} -8.26309 q^{87} -8.63142 q^{89} +4.72205i q^{91} +0.742978i q^{93} +3.44460 q^{95} -0.755814 q^{97} +3.79816i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{7} - 16 q^{9} + 18 q^{15} + 6 q^{17} + 4 q^{23} - 20 q^{25} + 44 q^{31} - 16 q^{33} + 2 q^{39} - 20 q^{41} + 10 q^{47} + 64 q^{49} + 16 q^{55} - 12 q^{57} + 88 q^{63} + 2 q^{65} + 10 q^{71}+ \cdots + 84 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 1.40634i − 0.811950i −0.913884 − 0.405975i \(-0.866932\pi\)
0.913884 − 0.405975i \(-0.133068\pi\)
\(4\) 0 0
\(5\) 0.422133i 0.188784i 0.995535 + 0.0943918i \(0.0300906\pi\)
−0.995535 + 0.0943918i \(0.969909\pi\)
\(6\) 0 0
\(7\) 4.72205 1.78477 0.892383 − 0.451278i \(-0.149032\pi\)
0.892383 + 0.451278i \(0.149032\pi\)
\(8\) 0 0
\(9\) 1.02221 0.340738
\(10\) 0 0
\(11\) 3.71563i 1.12030i 0.828390 + 0.560152i \(0.189257\pi\)
−0.828390 + 0.560152i \(0.810743\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 0.593662 0.153283
\(16\) 0 0
\(17\) 2.39054 0.579792 0.289896 − 0.957058i \(-0.406379\pi\)
0.289896 + 0.957058i \(0.406379\pi\)
\(18\) 0 0
\(19\) − 8.15998i − 1.87203i −0.351964 − 0.936013i \(-0.614486\pi\)
0.351964 − 0.936013i \(-0.385514\pi\)
\(20\) 0 0
\(21\) − 6.64080i − 1.44914i
\(22\) 0 0
\(23\) −7.87561 −1.64218 −0.821089 − 0.570801i \(-0.806633\pi\)
−0.821089 + 0.570801i \(0.806633\pi\)
\(24\) 0 0
\(25\) 4.82180 0.964361
\(26\) 0 0
\(27\) − 5.65659i − 1.08861i
\(28\) 0 0
\(29\) − 5.87561i − 1.09107i −0.838087 − 0.545536i \(-0.816326\pi\)
0.838087 − 0.545536i \(-0.183674\pi\)
\(30\) 0 0
\(31\) −0.528307 −0.0948867 −0.0474433 − 0.998874i \(-0.515107\pi\)
−0.0474433 + 0.998874i \(0.515107\pi\)
\(32\) 0 0
\(33\) 5.22543 0.909631
\(34\) 0 0
\(35\) 1.99333i 0.336935i
\(36\) 0 0
\(37\) − 1.20954i − 0.198847i −0.995045 − 0.0994233i \(-0.968300\pi\)
0.995045 − 0.0994233i \(-0.0316998\pi\)
\(38\) 0 0
\(39\) 1.40634 0.225194
\(40\) 0 0
\(41\) −9.03134 −1.41046 −0.705229 − 0.708979i \(-0.749156\pi\)
−0.705229 + 0.708979i \(0.749156\pi\)
\(42\) 0 0
\(43\) 2.19374i 0.334542i 0.985911 + 0.167271i \(0.0534956\pi\)
−0.985911 + 0.167271i \(0.946504\pi\)
\(44\) 0 0
\(45\) 0.431510i 0.0643257i
\(46\) 0 0
\(47\) 11.1348 1.62418 0.812089 − 0.583533i \(-0.198330\pi\)
0.812089 + 0.583533i \(0.198330\pi\)
\(48\) 0 0
\(49\) 15.2977 2.18539
\(50\) 0 0
\(51\) − 3.36191i − 0.470762i
\(52\) 0 0
\(53\) 5.03134i 0.691108i 0.938399 + 0.345554i \(0.112309\pi\)
−0.938399 + 0.345554i \(0.887691\pi\)
\(54\) 0 0
\(55\) −1.56849 −0.211495
\(56\) 0 0
\(57\) −11.4757 −1.51999
\(58\) 0 0
\(59\) 3.71563i 0.483734i 0.970309 + 0.241867i \(0.0777597\pi\)
−0.970309 + 0.241867i \(0.922240\pi\)
\(60\) 0 0
\(61\) 14.6567i 1.87660i 0.345826 + 0.938299i \(0.387599\pi\)
−0.345826 + 0.938299i \(0.612401\pi\)
\(62\) 0 0
\(63\) 4.82694 0.608137
\(64\) 0 0
\(65\) −0.422133 −0.0523592
\(66\) 0 0
\(67\) − 6.47144i − 0.790613i −0.918549 − 0.395306i \(-0.870638\pi\)
0.918549 − 0.395306i \(-0.129362\pi\)
\(68\) 0 0
\(69\) 11.0758i 1.33337i
\(70\) 0 0
\(71\) 9.34063 1.10853 0.554265 − 0.832341i \(-0.313000\pi\)
0.554265 + 0.832341i \(0.313000\pi\)
\(72\) 0 0
\(73\) 16.5070 1.93200 0.966001 − 0.258540i \(-0.0832413\pi\)
0.966001 + 0.258540i \(0.0832413\pi\)
\(74\) 0 0
\(75\) − 6.78109i − 0.783012i
\(76\) 0 0
\(77\) 17.5454i 1.99948i
\(78\) 0 0
\(79\) 11.4313 1.28612 0.643059 − 0.765817i \(-0.277665\pi\)
0.643059 + 0.765817i \(0.277665\pi\)
\(80\) 0 0
\(81\) −4.88844 −0.543160
\(82\) 0 0
\(83\) − 6.08396i − 0.667801i −0.942608 − 0.333901i \(-0.891635\pi\)
0.942608 − 0.333901i \(-0.108365\pi\)
\(84\) 0 0
\(85\) 1.00913i 0.109455i
\(86\) 0 0
\(87\) −8.26309 −0.885896
\(88\) 0 0
\(89\) −8.63142 −0.914929 −0.457464 − 0.889228i \(-0.651242\pi\)
−0.457464 + 0.889228i \(0.651242\pi\)
\(90\) 0 0
\(91\) 4.72205i 0.495005i
\(92\) 0 0
\(93\) 0.742978i 0.0770432i
\(94\) 0 0
\(95\) 3.44460 0.353408
\(96\) 0 0
\(97\) −0.755814 −0.0767413 −0.0383706 − 0.999264i \(-0.512217\pi\)
−0.0383706 + 0.999264i \(0.512217\pi\)
\(98\) 0 0
\(99\) 3.79816i 0.381730i
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 3328.2.b.bc.1665.3 10
4.3 odd 2 3328.2.b.bd.1665.8 10
8.3 odd 2 3328.2.b.bd.1665.3 10
8.5 even 2 inner 3328.2.b.bc.1665.8 10
16.3 odd 4 1664.2.a.y.1.2 ✓ 5
16.5 even 4 1664.2.a.z.1.2 yes 5
16.11 odd 4 1664.2.a.bb.1.4 yes 5
16.13 even 4 1664.2.a.ba.1.4 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.a.y.1.2 ✓ 5 16.3 odd 4
1664.2.a.z.1.2 yes 5 16.5 even 4
1664.2.a.ba.1.4 yes 5 16.13 even 4
1664.2.a.bb.1.4 yes 5 16.11 odd 4
3328.2.b.bc.1665.3 10 1.1 even 1 trivial
3328.2.b.bc.1665.8 10 8.5 even 2 inner
3328.2.b.bd.1665.3 10 8.3 odd 2
3328.2.b.bd.1665.8 10 4.3 odd 2