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Properties

Label 9200.2.a.bw.1.2
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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 = [2,0,1,0,0,0,1,0,3,0,2] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 1150)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) 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)
 
\(f(q)\) \(=\) \(q+2.56155 q^{3} -1.56155 q^{7} +3.56155 q^{9} +1.00000 q^{11} -3.56155 q^{13} -1.43845 q^{17} -3.00000 q^{19} -4.00000 q^{21} +1.00000 q^{23} +1.43845 q^{27} +5.56155 q^{29} -3.12311 q^{31} +2.56155 q^{33} +5.12311 q^{37} -9.12311 q^{39} -10.1231 q^{41} -4.68466 q^{43} +6.00000 q^{47} -4.56155 q^{49} -3.68466 q^{51} -0.876894 q^{53} -7.68466 q^{57} -4.00000 q^{59} -14.2462 q^{61} -5.56155 q^{63} -1.43845 q^{67} +2.56155 q^{69} +7.12311 q^{71} -2.12311 q^{73} -1.56155 q^{77} +12.9309 q^{79} -7.00000 q^{81} -2.12311 q^{83} +14.2462 q^{87} -12.5616 q^{89} +5.56155 q^{91} -8.00000 q^{93} +8.24621 q^{97} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 7 q^{17} - 6 q^{19} - 8 q^{21} + 2 q^{23} + 7 q^{27} + 7 q^{29} + 2 q^{31} + q^{33} + 2 q^{37} - 10 q^{39} - 12 q^{41} + 3 q^{43} + 12 q^{47} - 5 q^{49}+ \cdots + 3 q^{99}+O(q^{100}) \) 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
\(3\) 2.56155 1.47891 0.739457 − 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.56155 −0.590211 −0.295106 − 0.955465i \(-0.595355\pi\)
−0.295106 + 0.955465i \(0.595355\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −3.56155 −0.987797 −0.493899 − 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.43845 −0.348875 −0.174437 − 0.984668i \(-0.555811\pi\)
−0.174437 + 0.984668i \(0.555811\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) 5.56155 1.03275 0.516377 − 0.856361i \(-0.327280\pi\)
0.516377 + 0.856361i \(0.327280\pi\)
\(30\) 0 0
\(31\) −3.12311 −0.560926 −0.280463 − 0.959865i \(-0.590488\pi\)
−0.280463 + 0.959865i \(0.590488\pi\)
\(32\) 0 0
\(33\) 2.56155 0.445909
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.12311 0.842233 0.421117 − 0.907006i \(-0.361638\pi\)
0.421117 + 0.907006i \(0.361638\pi\)
\(38\) 0 0
\(39\) −9.12311 −1.46087
\(40\) 0 0
\(41\) −10.1231 −1.58096 −0.790482 − 0.612486i \(-0.790170\pi\)
−0.790482 + 0.612486i \(0.790170\pi\)
\(42\) 0 0
\(43\) −4.68466 −0.714404 −0.357202 − 0.934027i \(-0.616269\pi\)
−0.357202 + 0.934027i \(0.616269\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) −3.68466 −0.515955
\(52\) 0 0
\(53\) −0.876894 −0.120451 −0.0602254 − 0.998185i \(-0.519182\pi\)
−0.0602254 + 0.998185i \(0.519182\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −7.68466 −1.01786
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −14.2462 −1.82404 −0.912020 − 0.410145i \(-0.865478\pi\)
−0.912020 + 0.410145i \(0.865478\pi\)
\(62\) 0 0
\(63\) −5.56155 −0.700690
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.43845 −0.175734 −0.0878671 − 0.996132i \(-0.528005\pi\)
−0.0878671 + 0.996132i \(0.528005\pi\)
\(68\) 0 0
\(69\) 2.56155 0.308375
\(70\) 0 0
\(71\) 7.12311 0.845357 0.422679 − 0.906280i \(-0.361090\pi\)
0.422679 + 0.906280i \(0.361090\pi\)
\(72\) 0 0
\(73\) −2.12311 −0.248491 −0.124245 − 0.992252i \(-0.539651\pi\)
−0.124245 + 0.992252i \(0.539651\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.56155 −0.177955
\(78\) 0 0
\(79\) 12.9309 1.45484 0.727418 − 0.686194i \(-0.240720\pi\)
0.727418 + 0.686194i \(0.240720\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −2.12311 −0.233041 −0.116521 − 0.993188i \(-0.537174\pi\)
−0.116521 + 0.993188i \(0.537174\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 14.2462 1.52735
\(88\) 0 0
\(89\) −12.5616 −1.33152 −0.665761 − 0.746165i \(-0.731893\pi\)
−0.665761 + 0.746165i \(0.731893\pi\)
\(90\) 0 0
\(91\) 5.56155 0.583009
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.24621 0.837276 0.418638 − 0.908153i \(-0.362508\pi\)
0.418638 + 0.908153i \(0.362508\pi\)
\(98\) 0 0
\(99\) 3.56155 0.357950
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.bw.1.2 2
4.3 odd 2 1150.2.a.k.1.1 ✓ 2
5.4 even 2 9200.2.a.bp.1.1 2
20.3 even 4 1150.2.b.h.599.3 4
20.7 even 4 1150.2.b.h.599.2 4
20.19 odd 2 1150.2.a.p.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.2.a.k.1.1 ✓ 2 4.3 odd 2
1150.2.a.p.1.2 yes 2 20.19 odd 2
1150.2.b.h.599.2 4 20.7 even 4
1150.2.b.h.599.3 4 20.3 even 4
9200.2.a.bp.1.1 2 5.4 even 2
9200.2.a.bw.1.2 2 1.1 even 1 trivial