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(1,1682)] 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("1682.1"); 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([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,6,-8,6,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.4308376200\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{28})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 7x^{4} + 14x^{2} - 7 \) 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)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.56366\) of defining polynomial
Character \(\chi\) \(=\) 1682.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.80194 q^{3} +1.00000 q^{4} +1.81762 q^{5} -2.80194 q^{6} +1.41574 q^{7} +1.00000 q^{8} +4.85086 q^{9} +1.81762 q^{10} -3.53747 q^{11} -2.80194 q^{12} -5.56849 q^{13} +1.41574 q^{14} -5.09287 q^{15} +1.00000 q^{16} -7.46337 q^{17} +4.85086 q^{18} +5.03677 q^{19} +1.81762 q^{20} -3.96683 q^{21} -3.53747 q^{22} +1.78936 q^{23} -2.80194 q^{24} -1.69625 q^{25} -5.56849 q^{26} -5.18598 q^{27} +1.41574 q^{28} -5.09287 q^{30} -4.78772 q^{31} +1.00000 q^{32} +9.91178 q^{33} -7.46337 q^{34} +2.57329 q^{35} +4.85086 q^{36} -1.07410 q^{37} +5.03677 q^{38} +15.6026 q^{39} +1.81762 q^{40} +1.71164 q^{41} -3.96683 q^{42} +2.03143 q^{43} -3.53747 q^{44} +8.81703 q^{45} +1.78936 q^{46} -0.387225 q^{47} -2.80194 q^{48} -4.99567 q^{49} -1.69625 q^{50} +20.9119 q^{51} -5.56849 q^{52} -2.69352 q^{53} -5.18598 q^{54} -6.42979 q^{55} +1.41574 q^{56} -14.1127 q^{57} -3.81302 q^{59} -5.09287 q^{60} -13.5048 q^{61} -4.78772 q^{62} +6.86757 q^{63} +1.00000 q^{64} -10.1214 q^{65} +9.91178 q^{66} -2.96157 q^{67} -7.46337 q^{68} -5.01367 q^{69} +2.57329 q^{70} +6.27857 q^{71} +4.85086 q^{72} -0.960361 q^{73} -1.07410 q^{74} +4.75277 q^{75} +5.03677 q^{76} -5.00816 q^{77} +15.6026 q^{78} +2.78205 q^{79} +1.81762 q^{80} -0.0217703 q^{81} +1.71164 q^{82} -9.07216 q^{83} -3.96683 q^{84} -13.5656 q^{85} +2.03143 q^{86} -3.53747 q^{88} +5.31049 q^{89} +8.81703 q^{90} -7.88357 q^{91} +1.78936 q^{92} +13.4149 q^{93} -0.387225 q^{94} +9.15496 q^{95} -2.80194 q^{96} -10.9945 q^{97} -4.99567 q^{98} -17.1598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - 8 q^{3} + 6 q^{4} - 6 q^{5} - 8 q^{6} + 2 q^{7} + 6 q^{8} + 2 q^{9} - 6 q^{10} - 2 q^{11} - 8 q^{12} - 8 q^{13} + 2 q^{14} + 8 q^{15} + 6 q^{16} - 12 q^{17} + 2 q^{18} - 4 q^{19} - 6 q^{20}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.80194 −1.61770 −0.808850 − 0.588015i \(-0.799909\pi\)
−0.808850 + 0.588015i \(0.799909\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.81762 0.812866 0.406433 − 0.913681i \(-0.366772\pi\)
0.406433 + 0.913681i \(0.366772\pi\)
\(6\) −2.80194 −1.14389
\(7\) 1.41574 0.535101 0.267551 − 0.963544i \(-0.413786\pi\)
0.267551 + 0.963544i \(0.413786\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.85086 1.61695
\(10\) 1.81762 0.574783
\(11\) −3.53747 −1.06659 −0.533294 − 0.845930i \(-0.679046\pi\)
−0.533294 + 0.845930i \(0.679046\pi\)
\(12\) −2.80194 −0.808850
\(13\) −5.56849 −1.54442 −0.772211 − 0.635366i \(-0.780849\pi\)
−0.772211 + 0.635366i \(0.780849\pi\)
\(14\) 1.41574 0.378374
\(15\) −5.09287 −1.31497
\(16\) 1.00000 0.250000
\(17\) −7.46337 −1.81013 −0.905067 − 0.425269i \(-0.860180\pi\)
−0.905067 + 0.425269i \(0.860180\pi\)
\(18\) 4.85086 1.14336
\(19\) 5.03677 1.15552 0.577758 − 0.816208i \(-0.303928\pi\)
0.577758 + 0.816208i \(0.303928\pi\)
\(20\) 1.81762 0.406433
\(21\) −3.96683 −0.865633
\(22\) −3.53747 −0.754192
\(23\) 1.78936 0.373107 0.186554 − 0.982445i \(-0.440268\pi\)
0.186554 + 0.982445i \(0.440268\pi\)
\(24\) −2.80194 −0.571943
\(25\) −1.69625 −0.339249
\(26\) −5.56849 −1.09207
\(27\) −5.18598 −0.998042
\(28\) 1.41574 0.267551
\(29\) 0 0
\(30\) −5.09287 −0.929826
\(31\) −4.78772 −0.859900 −0.429950 − 0.902853i \(-0.641469\pi\)
−0.429950 + 0.902853i \(0.641469\pi\)
\(32\) 1.00000 0.176777
\(33\) 9.91178 1.72542
\(34\) −7.46337 −1.27996
\(35\) 2.57329 0.434966
\(36\) 4.85086 0.808476
\(37\) −1.07410 −0.176581 −0.0882903 − 0.996095i \(-0.528140\pi\)
−0.0882903 + 0.996095i \(0.528140\pi\)
\(38\) 5.03677 0.817073
\(39\) 15.6026 2.49841
\(40\) 1.81762 0.287391
\(41\) 1.71164 0.267314 0.133657 − 0.991028i \(-0.457328\pi\)
0.133657 + 0.991028i \(0.457328\pi\)
\(42\) −3.96683 −0.612095
\(43\) 2.03143 0.309790 0.154895 − 0.987931i \(-0.450496\pi\)
0.154895 + 0.987931i \(0.450496\pi\)
\(44\) −3.53747 −0.533294
\(45\) 8.81703 1.31436
\(46\) 1.78936 0.263827
\(47\) −0.387225 −0.0564826 −0.0282413 − 0.999601i \(-0.508991\pi\)
−0.0282413 + 0.999601i \(0.508991\pi\)
\(48\) −2.80194 −0.404425
\(49\) −4.99567 −0.713667
\(50\) −1.69625 −0.239885
\(51\) 20.9119 2.92825
\(52\) −5.56849 −0.772211
\(53\) −2.69352 −0.369983 −0.184992 − 0.982740i \(-0.559226\pi\)
−0.184992 + 0.982740i \(0.559226\pi\)
\(54\) −5.18598 −0.705723
\(55\) −6.42979 −0.866993
\(56\) 1.41574 0.189187
\(57\) −14.1127 −1.86928
\(58\) 0 0
\(59\) −3.81302 −0.496413 −0.248206 − 0.968707i \(-0.579841\pi\)
−0.248206 + 0.968707i \(0.579841\pi\)
\(60\) −5.09287 −0.657486
\(61\) −13.5048 −1.72911 −0.864554 − 0.502540i \(-0.832399\pi\)
−0.864554 + 0.502540i \(0.832399\pi\)
\(62\) −4.78772 −0.608041
\(63\) 6.86757 0.865233
\(64\) 1.00000 0.125000
\(65\) −10.1214 −1.25541
\(66\) 9.91178 1.22006
\(67\) −2.96157 −0.361814 −0.180907 − 0.983500i \(-0.557903\pi\)
−0.180907 + 0.983500i \(0.557903\pi\)
\(68\) −7.46337 −0.905067
\(69\) −5.01367 −0.603575
\(70\) 2.57329 0.307567
\(71\) 6.27857 0.745130 0.372565 − 0.928006i \(-0.378478\pi\)
0.372565 + 0.928006i \(0.378478\pi\)
\(72\) 4.85086 0.571679
\(73\) −0.960361 −0.112402 −0.0562008 − 0.998419i \(-0.517899\pi\)
−0.0562008 + 0.998419i \(0.517899\pi\)
\(74\) −1.07410 −0.124861
\(75\) 4.75277 0.548803
\(76\) 5.03677 0.577758
\(77\) −5.00816 −0.570733
\(78\) 15.6026 1.76664
\(79\) 2.78205 0.313005 0.156502 − 0.987678i \(-0.449978\pi\)
0.156502 + 0.987678i \(0.449978\pi\)
\(80\) 1.81762 0.203216
\(81\) −0.0217703 −0.00241892
\(82\) 1.71164 0.189019
\(83\) −9.07216 −0.995799 −0.497899 − 0.867235i \(-0.665895\pi\)
−0.497899 + 0.867235i \(0.665895\pi\)
\(84\) −3.96683 −0.432817
\(85\) −13.5656 −1.47140
\(86\) 2.03143 0.219055
\(87\) 0 0
\(88\) −3.53747 −0.377096
\(89\) 5.31049 0.562911 0.281455 − 0.959574i \(-0.409183\pi\)
0.281455 + 0.959574i \(0.409183\pi\)
\(90\) 8.81703 0.929396
\(91\) −7.88357 −0.826422
\(92\) 1.78936 0.186554
\(93\) 13.4149 1.39106
\(94\) −0.387225 −0.0399393
\(95\) 9.15496 0.939279
\(96\) −2.80194 −0.285972
\(97\) −10.9945 −1.11632 −0.558162 − 0.829732i \(-0.688493\pi\)
−0.558162 + 0.829732i \(0.688493\pi\)
\(98\) −4.99567 −0.504639
\(99\) −17.1598 −1.72462
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.a.s.1.2 6
29.3 odd 28 58.2.e.a.9.2 ✓ 12
29.10 odd 28 58.2.e.a.13.2 yes 12
29.12 odd 4 1682.2.b.j.1681.7 12
29.17 odd 4 1682.2.b.j.1681.5 12
29.28 even 2 1682.2.a.r.1.6 6
87.32 even 28 522.2.n.a.415.1 12
87.68 even 28 522.2.n.a.361.1 12
116.3 even 28 464.2.y.c.241.2 12
116.39 even 28 464.2.y.c.129.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.9.2 ✓ 12 29.3 odd 28
58.2.e.a.13.2 yes 12 29.10 odd 28
464.2.y.c.129.2 12 116.39 even 28
464.2.y.c.241.2 12 116.3 even 28
522.2.n.a.361.1 12 87.68 even 28
522.2.n.a.415.1 12 87.32 even 28
1682.2.a.r.1.6 6 29.28 even 2
1682.2.a.s.1.2 6 1.1 even 1 trivial
1682.2.b.j.1681.5 12 29.17 odd 4
1682.2.b.j.1681.7 12 29.12 odd 4