Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2166,2,Mod(1,2166)] 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("2166.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2166, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,3,3,0,3,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.2955970778\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.22668 q^{5} +1.00000 q^{6} -0.652704 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.22668 q^{10} -1.53209 q^{11} +1.00000 q^{12} +0.467911 q^{13} -0.652704 q^{14} +2.22668 q^{15} +1.00000 q^{16} -2.10607 q^{17} +1.00000 q^{18} +2.22668 q^{20} -0.652704 q^{21} -1.53209 q^{22} +5.90167 q^{23} +1.00000 q^{24} -0.0418891 q^{25} +0.467911 q^{26} +1.00000 q^{27} -0.652704 q^{28} +8.33275 q^{29} +2.22668 q^{30} +8.63816 q^{31} +1.00000 q^{32} -1.53209 q^{33} -2.10607 q^{34} -1.45336 q^{35} +1.00000 q^{36} +4.67499 q^{37} +0.467911 q^{39} +2.22668 q^{40} -3.47565 q^{41} -0.652704 q^{42} +10.2909 q^{43} -1.53209 q^{44} +2.22668 q^{45} +5.90167 q^{46} -4.63816 q^{47} +1.00000 q^{48} -6.57398 q^{49} -0.0418891 q^{50} -2.10607 q^{51} +0.467911 q^{52} +11.9513 q^{53} +1.00000 q^{54} -3.41147 q^{55} -0.652704 q^{56} +8.33275 q^{58} -14.2344 q^{59} +2.22668 q^{60} -9.12061 q^{61} +8.63816 q^{62} -0.652704 q^{63} +1.00000 q^{64} +1.04189 q^{65} -1.53209 q^{66} -0.248970 q^{67} -2.10607 q^{68} +5.90167 q^{69} -1.45336 q^{70} +4.44831 q^{71} +1.00000 q^{72} -9.09152 q^{73} +4.67499 q^{74} -0.0418891 q^{75} +1.00000 q^{77} +0.467911 q^{78} -12.0574 q^{79} +2.22668 q^{80} +1.00000 q^{81} -3.47565 q^{82} +3.70914 q^{83} -0.652704 q^{84} -4.68954 q^{85} +10.2909 q^{86} +8.33275 q^{87} -1.53209 q^{88} -16.7665 q^{89} +2.22668 q^{90} -0.305407 q^{91} +5.90167 q^{92} +8.63816 q^{93} -4.63816 q^{94} +1.00000 q^{96} -2.46791 q^{97} -6.57398 q^{98} -1.53209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{12} + 6 q^{13} - 3 q^{14} + 3 q^{16} + 6 q^{17} + 3 q^{18} - 3 q^{21} + 6 q^{23} + 3 q^{24} + 3 q^{25} + 6 q^{26} + 3 q^{27}+ \cdots - 12 q^{98}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.22668 0.995802 0.497901 − 0.867234i \(-0.334104\pi\)
0.497901 + 0.867234i \(0.334104\pi\)
\(6\) 1.00000 0.408248
\(7\) −0.652704 −0.246699 −0.123349 − 0.992363i \(-0.539364\pi\)
−0.123349 + 0.992363i \(0.539364\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.22668 0.704139
\(11\) −1.53209 −0.461942 −0.230971 − 0.972961i \(-0.574190\pi\)
−0.230971 + 0.972961i \(0.574190\pi\)
\(12\) 1.00000 0.288675
\(13\) 0.467911 0.129775 0.0648876 − 0.997893i \(-0.479331\pi\)
0.0648876 + 0.997893i \(0.479331\pi\)
\(14\) −0.652704 −0.174442
\(15\) 2.22668 0.574927
\(16\) 1.00000 0.250000
\(17\) −2.10607 −0.510796 −0.255398 − 0.966836i \(-0.582207\pi\)
−0.255398 + 0.966836i \(0.582207\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0
\(20\) 2.22668 0.497901
\(21\) −0.652704 −0.142432
\(22\) −1.53209 −0.326642
\(23\) 5.90167 1.23058 0.615292 − 0.788299i \(-0.289038\pi\)
0.615292 + 0.788299i \(0.289038\pi\)
\(24\) 1.00000 0.204124
\(25\) −0.0418891 −0.00837781
\(26\) 0.467911 0.0917649
\(27\) 1.00000 0.192450
\(28\) −0.652704 −0.123349
\(29\) 8.33275 1.54735 0.773676 − 0.633581i \(-0.218416\pi\)
0.773676 + 0.633581i \(0.218416\pi\)
\(30\) 2.22668 0.406535
\(31\) 8.63816 1.55146 0.775729 − 0.631066i \(-0.217382\pi\)
0.775729 + 0.631066i \(0.217382\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.53209 −0.266702
\(34\) −2.10607 −0.361187
\(35\) −1.45336 −0.245663
\(36\) 1.00000 0.166667
\(37\) 4.67499 0.768564 0.384282 − 0.923216i \(-0.374449\pi\)
0.384282 + 0.923216i \(0.374449\pi\)
\(38\) 0 0
\(39\) 0.467911 0.0749257
\(40\) 2.22668 0.352069
\(41\) −3.47565 −0.542806 −0.271403 − 0.962466i \(-0.587488\pi\)
−0.271403 + 0.962466i \(0.587488\pi\)
\(42\) −0.652704 −0.100714
\(43\) 10.2909 1.56934 0.784671 − 0.619913i \(-0.212832\pi\)
0.784671 + 0.619913i \(0.212832\pi\)
\(44\) −1.53209 −0.230971
\(45\) 2.22668 0.331934
\(46\) 5.90167 0.870154
\(47\) −4.63816 −0.676545 −0.338272 − 0.941048i \(-0.609842\pi\)
−0.338272 + 0.941048i \(0.609842\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.57398 −0.939140
\(50\) −0.0418891 −0.00592401
\(51\) −2.10607 −0.294908
\(52\) 0.467911 0.0648876
\(53\) 11.9513 1.64164 0.820819 − 0.571189i \(-0.193517\pi\)
0.820819 + 0.571189i \(0.193517\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.41147 −0.460003
\(56\) −0.652704 −0.0872212
\(57\) 0 0
\(58\) 8.33275 1.09414
\(59\) −14.2344 −1.85316 −0.926582 − 0.376093i \(-0.877267\pi\)
−0.926582 + 0.376093i \(0.877267\pi\)
\(60\) 2.22668 0.287463
\(61\) −9.12061 −1.16778 −0.583888 − 0.811835i \(-0.698469\pi\)
−0.583888 + 0.811835i \(0.698469\pi\)
\(62\) 8.63816 1.09705
\(63\) −0.652704 −0.0822329
\(64\) 1.00000 0.125000
\(65\) 1.04189 0.129230
\(66\) −1.53209 −0.188587
\(67\) −0.248970 −0.0304166 −0.0152083 − 0.999884i \(-0.504841\pi\)
−0.0152083 + 0.999884i \(0.504841\pi\)
\(68\) −2.10607 −0.255398
\(69\) 5.90167 0.710478
\(70\) −1.45336 −0.173710
\(71\) 4.44831 0.527917 0.263959 − 0.964534i \(-0.414972\pi\)
0.263959 + 0.964534i \(0.414972\pi\)
\(72\) 1.00000 0.117851
\(73\) −9.09152 −1.06408 −0.532041 − 0.846719i \(-0.678575\pi\)
−0.532041 + 0.846719i \(0.678575\pi\)
\(74\) 4.67499 0.543457
\(75\) −0.0418891 −0.00483693
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0.467911 0.0529805
\(79\) −12.0574 −1.35656 −0.678280 − 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(80\) 2.22668 0.248951
\(81\) 1.00000 0.111111
\(82\) −3.47565 −0.383822
\(83\) 3.70914 0.407131 0.203566 − 0.979061i \(-0.434747\pi\)
0.203566 + 0.979061i \(0.434747\pi\)
\(84\) −0.652704 −0.0712158
\(85\) −4.68954 −0.508652
\(86\) 10.2909 1.10969
\(87\) 8.33275 0.893364
\(88\) −1.53209 −0.163321
\(89\) −16.7665 −1.77725 −0.888623 − 0.458638i \(-0.848338\pi\)
−0.888623 + 0.458638i \(0.848338\pi\)
\(90\) 2.22668 0.234713
\(91\) −0.305407 −0.0320154
\(92\) 5.90167 0.615292
\(93\) 8.63816 0.895735
\(94\) −4.63816 −0.478389
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −2.46791 −0.250578 −0.125289 − 0.992120i \(-0.539986\pi\)
−0.125289 + 0.992120i \(0.539986\pi\)
\(98\) −6.57398 −0.664072
\(99\) −1.53209 −0.153981
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 2166.2.a.u.1.3 3
3.2 odd 2 6498.2.a.bn.1.1 3
19.14 odd 18 114.2.i.d.25.1 ✓ 6
19.15 odd 18 114.2.i.d.73.1 yes 6
19.18 odd 2 2166.2.a.o.1.3 3
57.14 even 18 342.2.u.a.253.1 6
57.53 even 18 342.2.u.a.73.1 6
57.56 even 2 6498.2.a.bs.1.1 3
76.15 even 18 912.2.bo.f.529.1 6
76.71 even 18 912.2.bo.f.481.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.25.1 ✓ 6 19.14 odd 18
114.2.i.d.73.1 yes 6 19.15 odd 18
342.2.u.a.73.1 6 57.53 even 18
342.2.u.a.253.1 6 57.14 even 18
912.2.bo.f.481.1 6 76.71 even 18
912.2.bo.f.529.1 6 76.15 even 18
2166.2.a.o.1.3 3 19.18 odd 2
2166.2.a.u.1.3 3 1.1 even 1 trivial
6498.2.a.bn.1.1 3 3.2 odd 2
6498.2.a.bs.1.1 3 57.56 even 2