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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 1665.10
Root \(-2.04204i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1665
Dual form 3328.2.b.bc.1665.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.21195i q^{3} -1.62268i q^{5} -3.82180 q^{7} -7.31660 q^{9} +6.08407i q^{11} +1.00000i q^{13} +5.21195 q^{15} -4.80122 q^{17} -0.144792i q^{19} -12.2754i q^{21} -2.22886 q^{23} +2.36693 q^{25} -13.8647i q^{27} -0.228864i q^{29} +6.33982 q^{31} -19.5417 q^{33} +6.20155i q^{35} -2.10729i q^{37} -3.21195 q^{39} -7.47421 q^{41} +0.518018i q^{43} +11.8725i q^{45} -12.9396 q^{47} +7.60619 q^{49} -15.4213i q^{51} +3.47421i q^{53} +9.87247 q^{55} +0.465065 q^{57} +6.08407i q^{59} -5.37357i q^{61} +27.9626 q^{63} +1.62268 q^{65} -6.63549i q^{67} -7.15899i q^{69} +14.7702 q^{71} +3.00915 q^{73} +7.60244i q^{75} -23.2521i q^{77} +16.1681 q^{79} +22.5829 q^{81} -9.59946i q^{83} +7.79082i q^{85} +0.735100 q^{87} -0.780285 q^{89} -3.82180i q^{91} +20.3632i q^{93} -0.234951 q^{95} +1.44858 q^{97} -44.5147i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{7} - 16 q^{9} + 18 q^{15} + 6 q^{17} + 4 q^{23} - 20 q^{25} + 44 q^{31} - 16 q^{33} + 2 q^{39} - 20 q^{41} + 10 q^{47} + 64 q^{49} + 16 q^{55} - 12 q^{57} + 88 q^{63} + 2 q^{65} + 10 q^{71}+ \cdots + 84 q^{95}+O(q^{100}) \) 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
\(3\) 3.21195i 1.85442i 0.374544 + 0.927209i \(0.377799\pi\)
−0.374544 + 0.927209i \(0.622201\pi\)
\(4\) 0 0
\(5\) − 1.62268i − 0.725682i −0.931851 − 0.362841i \(-0.881807\pi\)
0.931851 − 0.362841i \(-0.118193\pi\)
\(6\) 0 0
\(7\) −3.82180 −1.44451 −0.722253 − 0.691629i \(-0.756893\pi\)
−0.722253 + 0.691629i \(0.756893\pi\)
\(8\) 0 0
\(9\) −7.31660 −2.43887
\(10\) 0 0
\(11\) 6.08407i 1.83442i 0.398408 + 0.917208i \(0.369563\pi\)
−0.398408 + 0.917208i \(0.630437\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 5.21195 1.34572
\(16\) 0 0
\(17\) −4.80122 −1.16447 −0.582233 − 0.813022i \(-0.697821\pi\)
−0.582233 + 0.813022i \(0.697821\pi\)
\(18\) 0 0
\(19\) − 0.144792i − 0.0332176i −0.999862 − 0.0166088i \(-0.994713\pi\)
0.999862 − 0.0166088i \(-0.00528699\pi\)
\(20\) 0 0
\(21\) − 12.2754i − 2.67872i
\(22\) 0 0
\(23\) −2.22886 −0.464750 −0.232375 − 0.972626i \(-0.574650\pi\)
−0.232375 + 0.972626i \(0.574650\pi\)
\(24\) 0 0
\(25\) 2.36693 0.473385
\(26\) 0 0
\(27\) − 13.8647i − 2.66826i
\(28\) 0 0
\(29\) − 0.228864i − 0.0424990i −0.999774 − 0.0212495i \(-0.993236\pi\)
0.999774 − 0.0212495i \(-0.00676444\pi\)
\(30\) 0 0
\(31\) 6.33982 1.13867 0.569333 − 0.822107i \(-0.307202\pi\)
0.569333 + 0.822107i \(0.307202\pi\)
\(32\) 0 0
\(33\) −19.5417 −3.40178
\(34\) 0 0
\(35\) 6.20155i 1.04825i
\(36\) 0 0
\(37\) − 2.10729i − 0.346436i −0.984883 − 0.173218i \(-0.944583\pi\)
0.984883 − 0.173218i \(-0.0554166\pi\)
\(38\) 0 0
\(39\) −3.21195 −0.514323
\(40\) 0 0
\(41\) −7.47421 −1.16728 −0.583638 − 0.812014i \(-0.698371\pi\)
−0.583638 + 0.812014i \(0.698371\pi\)
\(42\) 0 0
\(43\) 0.518018i 0.0789970i 0.999220 + 0.0394985i \(0.0125760\pi\)
−0.999220 + 0.0394985i \(0.987424\pi\)
\(44\) 0 0
\(45\) 11.8725i 1.76984i
\(46\) 0 0
\(47\) −12.9396 −1.88744 −0.943719 − 0.330747i \(-0.892699\pi\)
−0.943719 + 0.330747i \(0.892699\pi\)
\(48\) 0 0
\(49\) 7.60619 1.08660
\(50\) 0 0
\(51\) − 15.4213i − 2.15941i
\(52\) 0 0
\(53\) 3.47421i 0.477220i 0.971115 + 0.238610i \(0.0766918\pi\)
−0.971115 + 0.238610i \(0.923308\pi\)
\(54\) 0 0
\(55\) 9.87247 1.33120
\(56\) 0 0
\(57\) 0.465065 0.0615994
\(58\) 0 0
\(59\) 6.08407i 0.792079i 0.918234 + 0.396039i \(0.129616\pi\)
−0.918234 + 0.396039i \(0.870384\pi\)
\(60\) 0 0
\(61\) − 5.37357i − 0.688016i −0.938967 − 0.344008i \(-0.888215\pi\)
0.938967 − 0.344008i \(-0.111785\pi\)
\(62\) 0 0
\(63\) 27.9626 3.52296
\(64\) 0 0
\(65\) 1.62268 0.201268
\(66\) 0 0
\(67\) − 6.63549i − 0.810655i −0.914172 − 0.405327i \(-0.867158\pi\)
0.914172 − 0.405327i \(-0.132842\pi\)
\(68\) 0 0
\(69\) − 7.15899i − 0.861842i
\(70\) 0 0
\(71\) 14.7702 1.75290 0.876452 − 0.481489i \(-0.159904\pi\)
0.876452 + 0.481489i \(0.159904\pi\)
\(72\) 0 0
\(73\) 3.00915 0.352194 0.176097 − 0.984373i \(-0.443653\pi\)
0.176097 + 0.984373i \(0.443653\pi\)
\(74\) 0 0
\(75\) 7.60244i 0.877854i
\(76\) 0 0
\(77\) − 23.2521i − 2.64983i
\(78\) 0 0
\(79\) 16.1681 1.81906 0.909529 − 0.415640i \(-0.136442\pi\)
0.909529 + 0.415640i \(0.136442\pi\)
\(80\) 0 0
\(81\) 22.5829 2.50921
\(82\) 0 0
\(83\) − 9.59946i − 1.05368i −0.849965 − 0.526839i \(-0.823377\pi\)
0.849965 − 0.526839i \(-0.176623\pi\)
\(84\) 0 0
\(85\) 7.79082i 0.845033i
\(86\) 0 0
\(87\) 0.735100 0.0788110
\(88\) 0 0
\(89\) −0.780285 −0.0827101 −0.0413550 − 0.999145i \(-0.513167\pi\)
−0.0413550 + 0.999145i \(0.513167\pi\)
\(90\) 0 0
\(91\) − 3.82180i − 0.400634i
\(92\) 0 0
\(93\) 20.3632i 2.11156i
\(94\) 0 0
\(95\) −0.234951 −0.0241054
\(96\) 0 0
\(97\) 1.44858 0.147081 0.0735404 − 0.997292i \(-0.476570\pi\)
0.0735404 + 0.997292i \(0.476570\pi\)
\(98\) 0 0
\(99\) − 44.5147i − 4.47390i
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.1665.10 10
4.3 odd 2 3328.2.b.bd.1665.1 10
8.3 odd 2 3328.2.b.bd.1665.10 10
8.5 even 2 inner 3328.2.b.bc.1665.1 10
16.3 odd 4 1664.2.a.y.1.5 ✓ 5
16.5 even 4 1664.2.a.z.1.5 yes 5
16.11 odd 4 1664.2.a.bb.1.1 yes 5
16.13 even 4 1664.2.a.ba.1.1 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.a.y.1.5 ✓ 5 16.3 odd 4
1664.2.a.z.1.5 yes 5 16.5 even 4
1664.2.a.ba.1.1 yes 5 16.13 even 4
1664.2.a.bb.1.1 yes 5 16.11 odd 4
3328.2.b.bc.1665.1 10 8.5 even 2 inner
3328.2.b.bc.1665.10 10 1.1 even 1 trivial
3328.2.b.bd.1665.1 10 4.3 odd 2
3328.2.b.bd.1665.10 10 8.3 odd 2