Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6800,2,Mod(1,6800)] 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("6800.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,0,0,2,0,6,0,-2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.2982733745\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 136)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 6800.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.23607 q^{3} +3.23607 q^{7} +7.47214 q^{9} -3.23607 q^{11} +4.47214 q^{13} -1.00000 q^{17} -2.47214 q^{19} -10.4721 q^{21} +3.23607 q^{23} -14.4721 q^{27} +2.00000 q^{29} +3.23607 q^{31} +10.4721 q^{33} -6.94427 q^{37} -14.4721 q^{39} +2.00000 q^{41} -10.4721 q^{43} -4.94427 q^{47} +3.47214 q^{49} +3.23607 q^{51} +2.00000 q^{53} +8.00000 q^{57} -5.52786 q^{59} -10.9443 q^{61} +24.1803 q^{63} -12.0000 q^{67} -10.4721 q^{69} -4.76393 q^{71} +2.94427 q^{73} -10.4721 q^{77} +1.70820 q^{79} +24.4164 q^{81} +10.4721 q^{83} -6.47214 q^{87} -16.4721 q^{89} +14.4721 q^{91} -10.4721 q^{93} -2.00000 q^{97} -24.1803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{7} + 6 q^{9} - 2 q^{11} - 2 q^{17} + 4 q^{19} - 12 q^{21} + 2 q^{23} - 20 q^{27} + 4 q^{29} + 2 q^{31} + 12 q^{33} + 4 q^{37} - 20 q^{39} + 4 q^{41} - 12 q^{43} + 8 q^{47} - 2 q^{49}+ \cdots - 26 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.23607 −1.86834 −0.934172 − 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.23607 1.22312 0.611559 − 0.791199i \(-0.290543\pi\)
0.611559 + 0.791199i \(0.290543\pi\)
\(8\) 0 0
\(9\) 7.47214 2.49071
\(10\) 0 0
\(11\) −3.23607 −0.975711 −0.487856 − 0.872924i \(-0.662221\pi\)
−0.487856 + 0.872924i \(0.662221\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 − 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −2.47214 −0.567147 −0.283573 − 0.958951i \(-0.591520\pi\)
−0.283573 + 0.958951i \(0.591520\pi\)
\(20\) 0 0
\(21\) −10.4721 −2.28521
\(22\) 0 0
\(23\) 3.23607 0.674767 0.337383 − 0.941367i \(-0.390458\pi\)
0.337383 + 0.941367i \(0.390458\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −14.4721 −2.78516
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 3.23607 0.581215 0.290607 − 0.956842i \(-0.406143\pi\)
0.290607 + 0.956842i \(0.406143\pi\)
\(32\) 0 0
\(33\) 10.4721 1.82296
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.94427 −1.14163 −0.570816 − 0.821078i \(-0.693373\pi\)
−0.570816 + 0.821078i \(0.693373\pi\)
\(38\) 0 0
\(39\) −14.4721 −2.31740
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −10.4721 −1.59699 −0.798493 − 0.602004i \(-0.794369\pi\)
−0.798493 + 0.602004i \(0.794369\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.94427 −0.721196 −0.360598 − 0.932721i \(-0.617427\pi\)
−0.360598 + 0.932721i \(0.617427\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) 0 0
\(51\) 3.23607 0.453140
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) −5.52786 −0.719667 −0.359833 − 0.933017i \(-0.617166\pi\)
−0.359833 + 0.933017i \(0.617166\pi\)
\(60\) 0 0
\(61\) −10.9443 −1.40127 −0.700635 − 0.713520i \(-0.747100\pi\)
−0.700635 + 0.713520i \(0.747100\pi\)
\(62\) 0 0
\(63\) 24.1803 3.04644
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) −10.4721 −1.26070
\(70\) 0 0
\(71\) −4.76393 −0.565375 −0.282687 − 0.959212i \(-0.591226\pi\)
−0.282687 + 0.959212i \(0.591226\pi\)
\(72\) 0 0
\(73\) 2.94427 0.344601 0.172300 − 0.985044i \(-0.444880\pi\)
0.172300 + 0.985044i \(0.444880\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.4721 −1.19341
\(78\) 0 0
\(79\) 1.70820 0.192188 0.0960940 − 0.995372i \(-0.469365\pi\)
0.0960940 + 0.995372i \(0.469365\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) 10.4721 1.14947 0.574733 − 0.818341i \(-0.305106\pi\)
0.574733 + 0.818341i \(0.305106\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.47214 −0.693886
\(88\) 0 0
\(89\) −16.4721 −1.74604 −0.873021 − 0.487682i \(-0.837843\pi\)
−0.873021 + 0.487682i \(0.837843\pi\)
\(90\) 0 0
\(91\) 14.4721 1.51709
\(92\) 0 0
\(93\) −10.4721 −1.08591
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −24.1803 −2.43022
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 6800.2.a.bd.1.1 2
4.3 odd 2 3400.2.a.i.1.2 2
5.4 even 2 272.2.a.f.1.2 2
15.14 odd 2 2448.2.a.u.1.1 2
20.3 even 4 3400.2.e.f.2449.4 4
20.7 even 4 3400.2.e.f.2449.1 4
20.19 odd 2 136.2.a.c.1.1 ✓ 2
40.19 odd 2 1088.2.a.s.1.2 2
40.29 even 2 1088.2.a.o.1.1 2
60.59 even 2 1224.2.a.i.1.2 2
85.84 even 2 4624.2.a.h.1.1 2
120.29 odd 2 9792.2.a.da.1.1 2
120.59 even 2 9792.2.a.db.1.2 2
140.139 even 2 6664.2.a.i.1.2 2
340.259 odd 4 2312.2.b.g.577.4 4
340.319 odd 4 2312.2.b.g.577.1 4
340.339 odd 2 2312.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.a.c.1.1 ✓ 2 20.19 odd 2
272.2.a.f.1.2 2 5.4 even 2
1088.2.a.o.1.1 2 40.29 even 2
1088.2.a.s.1.2 2 40.19 odd 2
1224.2.a.i.1.2 2 60.59 even 2
2312.2.a.m.1.2 2 340.339 odd 2
2312.2.b.g.577.1 4 340.319 odd 4
2312.2.b.g.577.4 4 340.259 odd 4
2448.2.a.u.1.1 2 15.14 odd 2
3400.2.a.i.1.2 2 4.3 odd 2
3400.2.e.f.2449.1 4 20.7 even 4
3400.2.e.f.2449.4 4 20.3 even 4
4624.2.a.h.1.1 2 85.84 even 2
6664.2.a.i.1.2 2 140.139 even 2
6800.2.a.bd.1.1 2 1.1 even 1 trivial
9792.2.a.da.1.1 2 120.29 odd 2
9792.2.a.db.1.2 2 120.59 even 2