Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9200,2,Mod(1,9200)] 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("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,0,0,3,0,-3,0,9] 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: \(73.4623698596\)
Analytic rank: \(0\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 9200.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.61803 q^{3} -1.85410 q^{7} -0.381966 q^{9} +5.61803 q^{11} -2.61803 q^{13} -0.854102 q^{17} +0.145898 q^{19} -3.00000 q^{21} +1.00000 q^{23} -5.47214 q^{27} -9.70820 q^{29} +2.14590 q^{31} +9.09017 q^{33} +9.70820 q^{37} -4.23607 q^{39} -5.61803 q^{41} +11.2361 q^{43} +1.70820 q^{47} -3.56231 q^{49} -1.38197 q^{51} +2.00000 q^{53} +0.236068 q^{57} +6.00000 q^{59} +2.85410 q^{61} +0.708204 q^{63} +5.23607 q^{67} +1.61803 q^{69} -0.381966 q^{71} +16.4721 q^{73} -10.4164 q^{77} +7.70820 q^{79} -7.70820 q^{81} +7.70820 q^{83} -15.7082 q^{87} +3.70820 q^{89} +4.85410 q^{91} +3.47214 q^{93} -13.0344 q^{97} -2.14590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 3 q^{7} - 3 q^{9} + 9 q^{11} - 3 q^{13} + 5 q^{17} + 7 q^{19} - 6 q^{21} + 2 q^{23} - 2 q^{27} - 6 q^{29} + 11 q^{31} + 7 q^{33} + 6 q^{37} - 4 q^{39} - 9 q^{41} + 18 q^{43} - 10 q^{47} + 13 q^{49}+ \cdots - 11 q^{99}+O(q^{100}) \) 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
\(3\) 1.61803 0.934172 0.467086 − 0.884212i \(-0.345304\pi\)
0.467086 + 0.884212i \(0.345304\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.85410 −0.700785 −0.350392 − 0.936603i \(-0.613952\pi\)
−0.350392 + 0.936603i \(0.613952\pi\)
\(8\) 0 0
\(9\) −0.381966 −0.127322
\(10\) 0 0
\(11\) 5.61803 1.69390 0.846950 − 0.531672i \(-0.178436\pi\)
0.846950 + 0.531672i \(0.178436\pi\)
\(12\) 0 0
\(13\) −2.61803 −0.726112 −0.363056 − 0.931767i \(-0.618267\pi\)
−0.363056 + 0.931767i \(0.618267\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.854102 −0.207150 −0.103575 − 0.994622i \(-0.533028\pi\)
−0.103575 + 0.994622i \(0.533028\pi\)
\(18\) 0 0
\(19\) 0.145898 0.0334713 0.0167357 − 0.999860i \(-0.494673\pi\)
0.0167357 + 0.999860i \(0.494673\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.47214 −1.05311
\(28\) 0 0
\(29\) −9.70820 −1.80277 −0.901384 − 0.433020i \(-0.857448\pi\)
−0.901384 + 0.433020i \(0.857448\pi\)
\(30\) 0 0
\(31\) 2.14590 0.385415 0.192707 − 0.981256i \(-0.438273\pi\)
0.192707 + 0.981256i \(0.438273\pi\)
\(32\) 0 0
\(33\) 9.09017 1.58240
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.70820 1.59602 0.798009 − 0.602645i \(-0.205886\pi\)
0.798009 + 0.602645i \(0.205886\pi\)
\(38\) 0 0
\(39\) −4.23607 −0.678314
\(40\) 0 0
\(41\) −5.61803 −0.877390 −0.438695 − 0.898636i \(-0.644559\pi\)
−0.438695 + 0.898636i \(0.644559\pi\)
\(42\) 0 0
\(43\) 11.2361 1.71348 0.856742 − 0.515745i \(-0.172485\pi\)
0.856742 + 0.515745i \(0.172485\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.70820 0.249167 0.124584 − 0.992209i \(-0.460241\pi\)
0.124584 + 0.992209i \(0.460241\pi\)
\(48\) 0 0
\(49\) −3.56231 −0.508901
\(50\) 0 0
\(51\) −1.38197 −0.193514
\(52\) 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
\(57\) 0.236068 0.0312680
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 2.85410 0.365430 0.182715 − 0.983166i \(-0.441511\pi\)
0.182715 + 0.983166i \(0.441511\pi\)
\(62\) 0 0
\(63\) 0.708204 0.0892253
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.23607 0.639688 0.319844 − 0.947470i \(-0.396370\pi\)
0.319844 + 0.947470i \(0.396370\pi\)
\(68\) 0 0
\(69\) 1.61803 0.194788
\(70\) 0 0
\(71\) −0.381966 −0.0453310 −0.0226655 − 0.999743i \(-0.507215\pi\)
−0.0226655 + 0.999743i \(0.507215\pi\)
\(72\) 0 0
\(73\) 16.4721 1.92792 0.963959 − 0.266051i \(-0.0857191\pi\)
0.963959 + 0.266051i \(0.0857191\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.4164 −1.18706
\(78\) 0 0
\(79\) 7.70820 0.867241 0.433620 − 0.901096i \(-0.357236\pi\)
0.433620 + 0.901096i \(0.357236\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) 0 0
\(83\) 7.70820 0.846085 0.423043 − 0.906110i \(-0.360962\pi\)
0.423043 + 0.906110i \(0.360962\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −15.7082 −1.68410
\(88\) 0 0
\(89\) 3.70820 0.393069 0.196534 − 0.980497i \(-0.437031\pi\)
0.196534 + 0.980497i \(0.437031\pi\)
\(90\) 0 0
\(91\) 4.85410 0.508848
\(92\) 0 0
\(93\) 3.47214 0.360044
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.0344 −1.32345 −0.661724 − 0.749748i \(-0.730175\pi\)
−0.661724 + 0.749748i \(0.730175\pi\)
\(98\) 0 0
\(99\) −2.14590 −0.215671
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 9200.2.a.by.1.2 2
4.3 odd 2 1150.2.a.n.1.1 2
5.2 odd 4 1840.2.e.c.369.1 4
5.3 odd 4 1840.2.e.c.369.4 4
5.4 even 2 9200.2.a.bo.1.1 2
20.3 even 4 230.2.b.a.139.1 ✓ 4
20.7 even 4 230.2.b.a.139.4 yes 4
20.19 odd 2 1150.2.a.l.1.2 2
60.23 odd 4 2070.2.d.c.829.4 4
60.47 odd 4 2070.2.d.c.829.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.b.a.139.1 ✓ 4 20.3 even 4
230.2.b.a.139.4 yes 4 20.7 even 4
1150.2.a.l.1.2 2 20.19 odd 2
1150.2.a.n.1.1 2 4.3 odd 2
1840.2.e.c.369.1 4 5.2 odd 4
1840.2.e.c.369.4 4 5.3 odd 4
2070.2.d.c.829.2 4 60.47 odd 4
2070.2.d.c.829.4 4 60.23 odd 4
9200.2.a.bo.1.1 2 5.4 even 2
9200.2.a.by.1.2 2 1.1 even 1 trivial