Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6498,2,Mod(1,6498)] 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("6498.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6498, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6498.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,0,3,0,0,-3,3,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(51.8867912334\)
Analytic rank: \(1\)
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 1.1
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 6498.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.22668 q^{5} -0.652704 q^{7} +1.00000 q^{8} -2.22668 q^{10} +1.53209 q^{11} -0.467911 q^{13} -0.652704 q^{14} +1.00000 q^{16} +2.10607 q^{17} -2.22668 q^{20} +1.53209 q^{22} -5.90167 q^{23} -0.0418891 q^{25} -0.467911 q^{26} -0.652704 q^{28} +8.33275 q^{29} -8.63816 q^{31} +1.00000 q^{32} +2.10607 q^{34} +1.45336 q^{35} -4.67499 q^{37} -2.22668 q^{40} -3.47565 q^{41} +10.2909 q^{43} +1.53209 q^{44} -5.90167 q^{46} +4.63816 q^{47} -6.57398 q^{49} -0.0418891 q^{50} -0.467911 q^{52} +11.9513 q^{53} -3.41147 q^{55} -0.652704 q^{56} +8.33275 q^{58} -14.2344 q^{59} -9.12061 q^{61} -8.63816 q^{62} +1.00000 q^{64} +1.04189 q^{65} +0.248970 q^{67} +2.10607 q^{68} +1.45336 q^{70} +4.44831 q^{71} -9.09152 q^{73} -4.67499 q^{74} -1.00000 q^{77} +12.0574 q^{79} -2.22668 q^{80} -3.47565 q^{82} -3.70914 q^{83} -4.68954 q^{85} +10.2909 q^{86} +1.53209 q^{88} -16.7665 q^{89} +0.305407 q^{91} -5.90167 q^{92} +4.63816 q^{94} +2.46791 q^{97} -6.57398 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{7} + 3 q^{8} - 6 q^{13} - 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{23} + 3 q^{25} - 6 q^{26} - 3 q^{28} + 6 q^{29} - 9 q^{31} + 3 q^{32} - 6 q^{34} - 9 q^{35} - 9 q^{37} + 9 q^{41}+ \cdots - 12 q^{98}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.22668 −0.995802 −0.497901 − 0.867234i \(-0.665896\pi\)
−0.497901 + 0.867234i \(0.665896\pi\)
\(6\) 0 0
\(7\) −0.652704 −0.246699 −0.123349 − 0.992363i \(-0.539364\pi\)
−0.123349 + 0.992363i \(0.539364\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.22668 −0.704139
\(11\) 1.53209 0.461942 0.230971 − 0.972961i \(-0.425810\pi\)
0.230971 + 0.972961i \(0.425810\pi\)
\(12\) 0 0
\(13\) −0.467911 −0.129775 −0.0648876 − 0.997893i \(-0.520669\pi\)
−0.0648876 + 0.997893i \(0.520669\pi\)
\(14\) −0.652704 −0.174442
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.10607 0.510796 0.255398 − 0.966836i \(-0.417793\pi\)
0.255398 + 0.966836i \(0.417793\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) −2.22668 −0.497901
\(21\) 0 0
\(22\) 1.53209 0.326642
\(23\) −5.90167 −1.23058 −0.615292 − 0.788299i \(-0.710962\pi\)
−0.615292 + 0.788299i \(0.710962\pi\)
\(24\) 0 0
\(25\) −0.0418891 −0.00837781
\(26\) −0.467911 −0.0917649
\(27\) 0 0
\(28\) −0.652704 −0.123349
\(29\) 8.33275 1.54735 0.773676 − 0.633581i \(-0.218416\pi\)
0.773676 + 0.633581i \(0.218416\pi\)
\(30\) 0 0
\(31\) −8.63816 −1.55146 −0.775729 − 0.631066i \(-0.782618\pi\)
−0.775729 + 0.631066i \(0.782618\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.10607 0.361187
\(35\) 1.45336 0.245663
\(36\) 0 0
\(37\) −4.67499 −0.768564 −0.384282 − 0.923216i \(-0.625551\pi\)
−0.384282 + 0.923216i \(0.625551\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −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 0
\(43\) 10.2909 1.56934 0.784671 − 0.619913i \(-0.212832\pi\)
0.784671 + 0.619913i \(0.212832\pi\)
\(44\) 1.53209 0.230971
\(45\) 0 0
\(46\) −5.90167 −0.870154
\(47\) 4.63816 0.676545 0.338272 − 0.941048i \(-0.390158\pi\)
0.338272 + 0.941048i \(0.390158\pi\)
\(48\) 0 0
\(49\) −6.57398 −0.939140
\(50\) −0.0418891 −0.00592401
\(51\) 0 0
\(52\) −0.467911 −0.0648876
\(53\) 11.9513 1.64164 0.820819 − 0.571189i \(-0.193517\pi\)
0.820819 + 0.571189i \(0.193517\pi\)
\(54\) 0 0
\(55\) −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 \(0.877267\pi\)
\(60\) 0 0
\(61\) −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 0
\(64\) 1.00000 0.125000
\(65\) 1.04189 0.129230
\(66\) 0 0
\(67\) 0.248970 0.0304166 0.0152083 − 0.999884i \(-0.495159\pi\)
0.0152083 + 0.999884i \(0.495159\pi\)
\(68\) 2.10607 0.255398
\(69\) 0 0
\(70\) 1.45336 0.173710
\(71\) 4.44831 0.527917 0.263959 − 0.964534i \(-0.414972\pi\)
0.263959 + 0.964534i \(0.414972\pi\)
\(72\) 0 0
\(73\) −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 0
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 12.0574 1.35656 0.678280 − 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(80\) −2.22668 −0.248951
\(81\) 0 0
\(82\) −3.47565 −0.383822
\(83\) −3.70914 −0.407131 −0.203566 − 0.979061i \(-0.565253\pi\)
−0.203566 + 0.979061i \(0.565253\pi\)
\(84\) 0 0
\(85\) −4.68954 −0.508652
\(86\) 10.2909 1.10969
\(87\) 0 0
\(88\) 1.53209 0.163321
\(89\) −16.7665 −1.77725 −0.888623 − 0.458638i \(-0.848338\pi\)
−0.888623 + 0.458638i \(0.848338\pi\)
\(90\) 0 0
\(91\) 0.305407 0.0320154
\(92\) −5.90167 −0.615292
\(93\) 0 0
\(94\) 4.63816 0.478389
\(95\) 0 0
\(96\) 0 0
\(97\) 2.46791 0.250578 0.125289 − 0.992120i \(-0.460014\pi\)
0.125289 + 0.992120i \(0.460014\pi\)
\(98\) −6.57398 −0.664072
\(99\) 0 0
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 6498.2.a.bs.1.1 3
3.2 odd 2 2166.2.a.o.1.3 3
19.4 even 9 342.2.u.a.73.1 6
19.5 even 9 342.2.u.a.253.1 6
19.18 odd 2 6498.2.a.bn.1.1 3
57.5 odd 18 114.2.i.d.25.1 ✓ 6
57.23 odd 18 114.2.i.d.73.1 yes 6
57.56 even 2 2166.2.a.u.1.3 3
228.23 even 18 912.2.bo.f.529.1 6
228.119 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 57.5 odd 18
114.2.i.d.73.1 yes 6 57.23 odd 18
342.2.u.a.73.1 6 19.4 even 9
342.2.u.a.253.1 6 19.5 even 9
912.2.bo.f.481.1 6 228.119 even 18
912.2.bo.f.529.1 6 228.23 even 18
2166.2.a.o.1.3 3 3.2 odd 2
2166.2.a.u.1.3 3 57.56 even 2
6498.2.a.bn.1.1 3 19.18 odd 2
6498.2.a.bs.1.1 3 1.1 even 1 trivial