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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 1665.4
Root \(-2.49707i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1665
Dual form 3328.2.b.bc.1665.7

$q$-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)
 
\(f(q)\) \(=\) \(q-0.738305i q^{3} +3.70891i q^{5} +1.43136 q^{7} +2.45491 q^{9} +2.99415i q^{11} -1.00000i q^{13} +2.73830 q^{15} +2.23230 q^{17} -1.24815i q^{19} -1.05678i q^{21} +8.24230 q^{23} -8.75600 q^{25} -4.02738i q^{27} -10.2423i q^{29} +10.4708 q^{31} +2.21059 q^{33} +5.30879i q^{35} +6.93152i q^{37} -0.738305 q^{39} -1.17552 q^{41} -9.90212i q^{43} +9.10502i q^{45} +9.11856 q^{47} -4.95121 q^{49} -1.64812i q^{51} +2.82448i q^{53} -11.1050 q^{55} -0.921515 q^{57} +2.99415i q^{59} +1.77770i q^{61} +3.51386 q^{63} +3.70891 q^{65} +13.5875i q^{67} -6.08533i q^{69} -3.08033 q^{71} -1.90297 q^{73} +6.46460i q^{75} +4.28571i q^{77} -1.98830 q^{79} +4.39128 q^{81} -2.21676i q^{83} +8.27939i q^{85} -7.56194 q^{87} -6.33933 q^{89} -1.43136i q^{91} -7.73061i q^{93} +4.62927 q^{95} -14.5816 q^{97} +7.35035i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{7} - 16 q^{9} + 18 q^{15} + 6 q^{17} + 4 q^{23} - 20 q^{25} + 44 q^{31} - 16 q^{33} + 2 q^{39} - 20 q^{41} + 10 q^{47} + 64 q^{49} + 16 q^{55} - 12 q^{57} + 88 q^{63} + 2 q^{65} + 10 q^{71}+ \cdots + 84 q^{95}+O(q^{100}) \) 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
\(3\) − 0.738305i − 0.426261i −0.977024 − 0.213130i \(-0.931634\pi\)
0.977024 − 0.213130i \(-0.0683659\pi\)
\(4\) 0 0
\(5\) 3.70891i 1.65867i 0.558749 + 0.829337i \(0.311282\pi\)
−0.558749 + 0.829337i \(0.688718\pi\)
\(6\) 0 0
\(7\) 1.43136 0.541004 0.270502 − 0.962719i \(-0.412810\pi\)
0.270502 + 0.962719i \(0.412810\pi\)
\(8\) 0 0
\(9\) 2.45491 0.818302
\(10\) 0 0
\(11\) 2.99415i 0.902770i 0.892329 + 0.451385i \(0.149070\pi\)
−0.892329 + 0.451385i \(0.850930\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) 0 0
\(15\) 2.73830 0.707027
\(16\) 0 0
\(17\) 2.23230 0.541412 0.270706 − 0.962662i \(-0.412743\pi\)
0.270706 + 0.962662i \(0.412743\pi\)
\(18\) 0 0
\(19\) − 1.24815i − 0.286345i −0.989698 − 0.143173i \(-0.954270\pi\)
0.989698 − 0.143173i \(-0.0457304\pi\)
\(20\) 0 0
\(21\) − 1.05678i − 0.230608i
\(22\) 0 0
\(23\) 8.24230 1.71864 0.859319 − 0.511440i \(-0.170888\pi\)
0.859319 + 0.511440i \(0.170888\pi\)
\(24\) 0 0
\(25\) −8.75600 −1.75120
\(26\) 0 0
\(27\) − 4.02738i − 0.775070i
\(28\) 0 0
\(29\) − 10.2423i − 1.90195i −0.309272 − 0.950974i \(-0.600085\pi\)
0.309272 − 0.950974i \(-0.399915\pi\)
\(30\) 0 0
\(31\) 10.4708 1.88060 0.940302 − 0.340342i \(-0.110543\pi\)
0.940302 + 0.340342i \(0.110543\pi\)
\(32\) 0 0
\(33\) 2.21059 0.384815
\(34\) 0 0
\(35\) 5.30879i 0.897349i
\(36\) 0 0
\(37\) 6.93152i 1.13953i 0.821806 + 0.569767i \(0.192967\pi\)
−0.821806 + 0.569767i \(0.807033\pi\)
\(38\) 0 0
\(39\) −0.738305 −0.118223
\(40\) 0 0
\(41\) −1.17552 −0.183585 −0.0917925 − 0.995778i \(-0.529260\pi\)
−0.0917925 + 0.995778i \(0.529260\pi\)
\(42\) 0 0
\(43\) − 9.90212i − 1.51006i −0.655691 − 0.755029i \(-0.727623\pi\)
0.655691 − 0.755029i \(-0.272377\pi\)
\(44\) 0 0
\(45\) 9.10502i 1.35730i
\(46\) 0 0
\(47\) 9.11856 1.33008 0.665040 − 0.746808i \(-0.268415\pi\)
0.665040 + 0.746808i \(0.268415\pi\)
\(48\) 0 0
\(49\) −4.95121 −0.707315
\(50\) 0 0
\(51\) − 1.64812i − 0.230782i
\(52\) 0 0
\(53\) 2.82448i 0.387973i 0.981004 + 0.193986i \(0.0621417\pi\)
−0.981004 + 0.193986i \(0.937858\pi\)
\(54\) 0 0
\(55\) −11.1050 −1.49740
\(56\) 0 0
\(57\) −0.921515 −0.122058
\(58\) 0 0
\(59\) 2.99415i 0.389805i 0.980823 + 0.194902i \(0.0624390\pi\)
−0.980823 + 0.194902i \(0.937561\pi\)
\(60\) 0 0
\(61\) 1.77770i 0.227611i 0.993503 + 0.113806i \(0.0363041\pi\)
−0.993503 + 0.113806i \(0.963696\pi\)
\(62\) 0 0
\(63\) 3.51386 0.442704
\(64\) 0 0
\(65\) 3.70891 0.460033
\(66\) 0 0
\(67\) 13.5875i 1.65998i 0.557782 + 0.829988i \(0.311653\pi\)
−0.557782 + 0.829988i \(0.688347\pi\)
\(68\) 0 0
\(69\) − 6.08533i − 0.732588i
\(70\) 0 0
\(71\) −3.08033 −0.365567 −0.182784 − 0.983153i \(-0.558511\pi\)
−0.182784 + 0.983153i \(0.558511\pi\)
\(72\) 0 0
\(73\) −1.90297 −0.222725 −0.111363 − 0.993780i \(-0.535522\pi\)
−0.111363 + 0.993780i \(0.535522\pi\)
\(74\) 0 0
\(75\) 6.46460i 0.746467i
\(76\) 0 0
\(77\) 4.28571i 0.488402i
\(78\) 0 0
\(79\) −1.98830 −0.223701 −0.111850 − 0.993725i \(-0.535678\pi\)
−0.111850 + 0.993725i \(0.535678\pi\)
\(80\) 0 0
\(81\) 4.39128 0.487920
\(82\) 0 0
\(83\) − 2.21676i − 0.243321i −0.992572 − 0.121660i \(-0.961178\pi\)
0.992572 − 0.121660i \(-0.0388218\pi\)
\(84\) 0 0
\(85\) 8.27939i 0.898026i
\(86\) 0 0
\(87\) −7.56194 −0.810725
\(88\) 0 0
\(89\) −6.33933 −0.671968 −0.335984 − 0.941868i \(-0.609069\pi\)
−0.335984 + 0.941868i \(0.609069\pi\)
\(90\) 0 0
\(91\) − 1.43136i − 0.150047i
\(92\) 0 0
\(93\) − 7.73061i − 0.801627i
\(94\) 0 0
\(95\) 4.62927 0.474954
\(96\) 0 0
\(97\) −14.5816 −1.48054 −0.740270 − 0.672310i \(-0.765302\pi\)
−0.740270 + 0.672310i \(0.765302\pi\)
\(98\) 0 0
\(99\) 7.35035i 0.738738i
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.1665.4 10
4.3 odd 2 3328.2.b.bd.1665.7 10
8.3 odd 2 3328.2.b.bd.1665.4 10
8.5 even 2 inner 3328.2.b.bc.1665.7 10
16.3 odd 4 1664.2.a.bb.1.2 yes 5
16.5 even 4 1664.2.a.ba.1.2 yes 5
16.11 odd 4 1664.2.a.y.1.4 ✓ 5
16.13 even 4 1664.2.a.z.1.4 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.a.y.1.4 ✓ 5 16.11 odd 4
1664.2.a.z.1.4 yes 5 16.13 even 4
1664.2.a.ba.1.2 yes 5 16.5 even 4
1664.2.a.bb.1.2 yes 5 16.3 odd 4
3328.2.b.bc.1665.4 10 1.1 even 1 trivial
3328.2.b.bc.1665.7 10 8.5 even 2 inner
3328.2.b.bd.1665.4 10 8.3 odd 2
3328.2.b.bd.1665.7 10 4.3 odd 2