Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1682,2,Mod(1681,1682)] 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("1682.1681"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1682, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-12,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4308376200\)
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: \( x^{12} + 30x^{10} + 341x^{8} + 1897x^{6} + 5456x^{4} + 7680x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1681.8
Root \(-1.74168i\) of defining polynomial
Character \(\chi\) \(=\) 1682.1681
Dual form 1682.2.b.i.1681.5

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.74168i q^{3} -1.00000 q^{4} +0.494698 q^{5} +1.74168 q^{6} +3.13840 q^{7} -1.00000i q^{8} -0.0334417 q^{9} +0.494698i q^{10} +1.39672i q^{11} +1.74168i q^{12} +5.71545 q^{13} +3.13840i q^{14} -0.861605i q^{15} +1.00000 q^{16} +5.31800i q^{17} -0.0334417i q^{18} +4.96656i q^{19} -0.494698 q^{20} -5.46607i q^{21} -1.39672 q^{22} -0.813609 q^{23} -1.74168 q^{24} -4.75527 q^{25} +5.71545i q^{26} -5.16679i q^{27} -3.13840 q^{28} +0.861605 q^{30} +6.36328i q^{31} +1.00000i q^{32} +2.43263 q^{33} -5.31800 q^{34} +1.55256 q^{35} +0.0334417 q^{36} -8.92766i q^{37} -4.96656 q^{38} -9.95448i q^{39} -0.494698i q^{40} +4.01226i q^{41} +5.46607 q^{42} -1.39672i q^{43} -1.39672i q^{44} -0.0165436 q^{45} -0.813609i q^{46} +10.2982i q^{47} -1.74168i q^{48} +2.84952 q^{49} -4.75527i q^{50} +9.26224 q^{51} -5.71545 q^{52} +4.24359 q^{53} +5.16679 q^{54} +0.690954i q^{55} -3.13840i q^{56} +8.65014 q^{57} +11.1598 q^{59} +0.861605i q^{60} +4.86648i q^{61} -6.36328 q^{62} -0.104953 q^{63} -1.00000 q^{64} +2.82742 q^{65} +2.43263i q^{66} -7.13840 q^{67} -5.31800i q^{68} +1.41705i q^{69} +1.55256i q^{70} -6.03736 q^{71} +0.0334417i q^{72} -12.2792i q^{73} +8.92766 q^{74} +8.28215i q^{75} -4.96656i q^{76} +4.38345i q^{77} +9.95448 q^{78} -9.40223i q^{79} +0.494698 q^{80} -9.09921 q^{81} -4.01226 q^{82} +12.7094 q^{83} +5.46607i q^{84} +2.63080i q^{85} +1.39672 q^{86} +1.39672 q^{88} +1.79217i q^{89} -0.0165436i q^{90} +17.9373 q^{91} +0.813609 q^{92} +11.0828 q^{93} -10.2982 q^{94} +2.45695i q^{95} +1.74168 q^{96} -3.79630i q^{97} +2.84952i q^{98} -0.0467086i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 4 q^{6} - 6 q^{7} - 24 q^{9} + 6 q^{13} + 12 q^{16} + 2 q^{22} - 28 q^{23} + 4 q^{24} + 8 q^{25} + 6 q^{28} + 54 q^{30} - 40 q^{33} + 12 q^{34} + 18 q^{35} + 24 q^{36} - 36 q^{38} + 20 q^{42}+ \cdots - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times\).

\(n\) \(843\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 1.74168i − 1.00556i −0.864415 − 0.502779i \(-0.832311\pi\)
0.864415 − 0.502779i \(-0.167689\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0.494698 0.221236 0.110618 − 0.993863i \(-0.464717\pi\)
0.110618 + 0.993863i \(0.464717\pi\)
\(6\) 1.74168 0.711037
\(7\) 3.13840 1.18620 0.593101 − 0.805128i \(-0.297904\pi\)
0.593101 + 0.805128i \(0.297904\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −0.0334417 −0.0111472
\(10\) 0.494698i 0.156437i
\(11\) 1.39672i 0.421126i 0.977580 + 0.210563i \(0.0675297\pi\)
−0.977580 + 0.210563i \(0.932470\pi\)
\(12\) 1.74168i 0.502779i
\(13\) 5.71545 1.58518 0.792591 − 0.609754i \(-0.208732\pi\)
0.792591 + 0.609754i \(0.208732\pi\)
\(14\) 3.13840i 0.838771i
\(15\) − 0.861605i − 0.222465i
\(16\) 1.00000 0.250000
\(17\) 5.31800i 1.28980i 0.764265 + 0.644902i \(0.223102\pi\)
−0.764265 + 0.644902i \(0.776898\pi\)
\(18\) − 0.0334417i − 0.00788229i
\(19\) 4.96656i 1.13941i 0.821850 + 0.569703i \(0.192942\pi\)
−0.821850 + 0.569703i \(0.807058\pi\)
\(20\) −0.494698 −0.110618
\(21\) − 5.46607i − 1.19279i
\(22\) −1.39672 −0.297781
\(23\) −0.813609 −0.169649 −0.0848246 − 0.996396i \(-0.527033\pi\)
−0.0848246 + 0.996396i \(0.527033\pi\)
\(24\) −1.74168 −0.355519
\(25\) −4.75527 −0.951055
\(26\) 5.71545i 1.12089i
\(27\) − 5.16679i − 0.994349i
\(28\) −3.13840 −0.593101
\(29\) 0 0
\(30\) 0.861605 0.157307
\(31\) 6.36328i 1.14288i 0.820645 + 0.571439i \(0.193615\pi\)
−0.820645 + 0.571439i \(0.806385\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 2.43263 0.423467
\(34\) −5.31800 −0.912029
\(35\) 1.55256 0.262430
\(36\) 0.0334417 0.00557362
\(37\) − 8.92766i − 1.46770i −0.679313 − 0.733849i \(-0.737722\pi\)
0.679313 − 0.733849i \(-0.262278\pi\)
\(38\) −4.96656 −0.805682
\(39\) − 9.95448i − 1.59399i
\(40\) − 0.494698i − 0.0782187i
\(41\) 4.01226i 0.626610i 0.949652 + 0.313305i \(0.101436\pi\)
−0.949652 + 0.313305i \(0.898564\pi\)
\(42\) 5.46607 0.843433
\(43\) − 1.39672i − 0.212997i −0.994313 − 0.106499i \(-0.966036\pi\)
0.994313 − 0.106499i \(-0.0339640\pi\)
\(44\) − 1.39672i − 0.210563i
\(45\) −0.0165436 −0.00246617
\(46\) − 0.813609i − 0.119960i
\(47\) 10.2982i 1.50214i 0.660223 + 0.751070i \(0.270462\pi\)
−0.660223 + 0.751070i \(0.729538\pi\)
\(48\) − 1.74168i − 0.251390i
\(49\) 2.84952 0.407075
\(50\) − 4.75527i − 0.672497i
\(51\) 9.26224 1.29697
\(52\) −5.71545 −0.792591
\(53\) 4.24359 0.582902 0.291451 − 0.956586i \(-0.405862\pi\)
0.291451 + 0.956586i \(0.405862\pi\)
\(54\) 5.16679 0.703111
\(55\) 0.690954i 0.0931682i
\(56\) − 3.13840i − 0.419386i
\(57\) 8.65014 1.14574
\(58\) 0 0
\(59\) 11.1598 1.45288 0.726438 − 0.687232i \(-0.241174\pi\)
0.726438 + 0.687232i \(0.241174\pi\)
\(60\) 0.861605i 0.111233i
\(61\) 4.86648i 0.623089i 0.950232 + 0.311544i \(0.100846\pi\)
−0.950232 + 0.311544i \(0.899154\pi\)
\(62\) −6.36328 −0.808137
\(63\) −0.104953 −0.0132229
\(64\) −1.00000 −0.125000
\(65\) 2.82742 0.350699
\(66\) 2.43263i 0.299436i
\(67\) −7.13840 −0.872094 −0.436047 − 0.899924i \(-0.643622\pi\)
−0.436047 + 0.899924i \(0.643622\pi\)
\(68\) − 5.31800i − 0.644902i
\(69\) 1.41705i 0.170592i
\(70\) 1.55256i 0.185566i
\(71\) −6.03736 −0.716502 −0.358251 − 0.933625i \(-0.616627\pi\)
−0.358251 + 0.933625i \(0.616627\pi\)
\(72\) 0.0334417i 0.00394114i
\(73\) − 12.2792i − 1.43717i −0.695439 − 0.718585i \(-0.744790\pi\)
0.695439 − 0.718585i \(-0.255210\pi\)
\(74\) 8.92766 1.03782
\(75\) 8.28215i 0.956341i
\(76\) − 4.96656i − 0.569703i
\(77\) 4.38345i 0.499541i
\(78\) 9.95448 1.12712
\(79\) − 9.40223i − 1.05783i −0.848674 − 0.528917i \(-0.822598\pi\)
0.848674 − 0.528917i \(-0.177402\pi\)
\(80\) 0.494698 0.0553089
\(81\) −9.09921 −1.01102
\(82\) −4.01226 −0.443080
\(83\) 12.7094 1.39504 0.697520 − 0.716565i \(-0.254287\pi\)
0.697520 + 0.716565i \(0.254287\pi\)
\(84\) 5.46607i 0.596397i
\(85\) 2.63080i 0.285351i
\(86\) 1.39672 0.150612
\(87\) 0 0
\(88\) 1.39672 0.148891
\(89\) 1.79217i 0.189970i 0.995479 + 0.0949849i \(0.0302803\pi\)
−0.995479 + 0.0949849i \(0.969720\pi\)
\(90\) − 0.0165436i − 0.00174384i
\(91\) 17.9373 1.88034
\(92\) 0.813609 0.0848246
\(93\) 11.0828 1.14923
\(94\) −10.2982 −1.06217
\(95\) 2.45695i 0.252078i
\(96\) 1.74168 0.177759
\(97\) − 3.79630i − 0.385456i −0.981252 − 0.192728i \(-0.938267\pi\)
0.981252 − 0.192728i \(-0.0617334\pi\)
\(98\) 2.84952i 0.287845i
\(99\) − 0.0467086i − 0.00469439i
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 1682.2.b.i.1681.8 12
29.2 odd 28 58.2.d.b.25.2 yes 12
29.12 odd 4 1682.2.a.q.1.5 6
29.14 odd 28 58.2.d.b.7.2 ✓ 12
29.17 odd 4 1682.2.a.t.1.2 6
29.28 even 2 inner 1682.2.b.i.1681.5 12
87.2 even 28 522.2.k.h.199.1 12
87.14 even 28 522.2.k.h.181.1 12
116.31 even 28 464.2.u.h.257.1 12
116.43 even 28 464.2.u.h.65.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.7.2 ✓ 12 29.14 odd 28
58.2.d.b.25.2 yes 12 29.2 odd 28
464.2.u.h.65.1 12 116.43 even 28
464.2.u.h.257.1 12 116.31 even 28
522.2.k.h.181.1 12 87.14 even 28
522.2.k.h.199.1 12 87.2 even 28
1682.2.a.q.1.5 6 29.12 odd 4
1682.2.a.t.1.2 6 29.17 odd 4
1682.2.b.i.1681.5 12 29.28 even 2 inner
1682.2.b.i.1681.8 12 1.1 even 1 trivial