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Properties

Label 9200.2.a.bw.1.1
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 1.1
Root \(-1.56155\) 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)
 
\(f(q)\) \(=\) \(q-1.56155 q^{3} +2.56155 q^{7} -0.561553 q^{9} +1.00000 q^{11} +0.561553 q^{13} -5.56155 q^{17} -3.00000 q^{19} -4.00000 q^{21} +1.00000 q^{23} +5.56155 q^{27} +1.43845 q^{29} +5.12311 q^{31} -1.56155 q^{33} -3.12311 q^{37} -0.876894 q^{39} -1.87689 q^{41} +7.68466 q^{43} +6.00000 q^{47} -0.438447 q^{49} +8.68466 q^{51} -9.12311 q^{53} +4.68466 q^{57} -4.00000 q^{59} +2.24621 q^{61} -1.43845 q^{63} -5.56155 q^{67} -1.56155 q^{69} -1.12311 q^{71} +6.12311 q^{73} +2.56155 q^{77} -15.9309 q^{79} -7.00000 q^{81} +6.12311 q^{83} -2.24621 q^{87} -8.43845 q^{89} +1.43845 q^{91} -8.00000 q^{93} -8.24621 q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 7 q^{17} - 6 q^{19} - 8 q^{21} + 2 q^{23} + 7 q^{27} + 7 q^{29} + 2 q^{31} + q^{33} + 2 q^{37} - 10 q^{39} - 12 q^{41} + 3 q^{43} + 12 q^{47} - 5 q^{49}+ \cdots + 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.56155 −0.901563 −0.450781 − 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.56155 0.968176 0.484088 − 0.875019i \(-0.339151\pi\)
0.484088 + 0.875019i \(0.339151\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 0.561553 0.155747 0.0778734 − 0.996963i \(-0.475187\pi\)
0.0778734 + 0.996963i \(0.475187\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.56155 −1.34887 −0.674437 − 0.738332i \(-0.735614\pi\)
−0.674437 + 0.738332i \(0.735614\pi\)
\(18\) 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
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) 1.43845 0.267113 0.133556 − 0.991041i \(-0.457360\pi\)
0.133556 + 0.991041i \(0.457360\pi\)
\(30\) 0 0
\(31\) 5.12311 0.920137 0.460068 − 0.887883i \(-0.347825\pi\)
0.460068 + 0.887883i \(0.347825\pi\)
\(32\) 0 0
\(33\) −1.56155 −0.271831
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.12311 −0.513435 −0.256718 − 0.966486i \(-0.582641\pi\)
−0.256718 + 0.966486i \(0.582641\pi\)
\(38\) 0 0
\(39\) −0.876894 −0.140415
\(40\) 0 0
\(41\) −1.87689 −0.293122 −0.146561 − 0.989202i \(-0.546820\pi\)
−0.146561 + 0.989202i \(0.546820\pi\)
\(42\) 0 0
\(43\) 7.68466 1.17190 0.585950 − 0.810347i \(-0.300722\pi\)
0.585950 + 0.810347i \(0.300722\pi\)
\(44\) 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
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) 8.68466 1.21610
\(52\) 0 0
\(53\) −9.12311 −1.25315 −0.626577 − 0.779359i \(-0.715545\pi\)
−0.626577 + 0.779359i \(0.715545\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.68466 0.620498
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.24621 0.287598 0.143799 − 0.989607i \(-0.454068\pi\)
0.143799 + 0.989607i \(0.454068\pi\)
\(62\) 0 0
\(63\) −1.43845 −0.181227
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.56155 −0.679452 −0.339726 − 0.940524i \(-0.610334\pi\)
−0.339726 + 0.940524i \(0.610334\pi\)
\(68\) 0 0
\(69\) −1.56155 −0.187989
\(70\) 0 0
\(71\) −1.12311 −0.133288 −0.0666441 − 0.997777i \(-0.521229\pi\)
−0.0666441 + 0.997777i \(0.521229\pi\)
\(72\) 0 0
\(73\) 6.12311 0.716655 0.358328 − 0.933596i \(-0.383347\pi\)
0.358328 + 0.933596i \(0.383347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.56155 0.291916
\(78\) 0 0
\(79\) −15.9309 −1.79236 −0.896181 − 0.443688i \(-0.853670\pi\)
−0.896181 + 0.443688i \(0.853670\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 6.12311 0.672098 0.336049 − 0.941844i \(-0.390909\pi\)
0.336049 + 0.941844i \(0.390909\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.24621 −0.240819
\(88\) 0 0
\(89\) −8.43845 −0.894474 −0.447237 − 0.894416i \(-0.647592\pi\)
−0.447237 + 0.894416i \(0.647592\pi\)
\(90\) 0 0
\(91\) 1.43845 0.150790
\(92\) 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 \(-0.637492\pi\)
−0.418638 + 0.908153i \(0.637492\pi\)
\(98\) 0 0
\(99\) −0.561553 −0.0564382
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.1.1 2
4.3 odd 2 1150.2.a.k.1.2 ✓ 2
5.4 even 2 9200.2.a.bp.1.2 2
20.3 even 4 1150.2.b.h.599.4 4
20.7 even 4 1150.2.b.h.599.1 4
20.19 odd 2 1150.2.a.p.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.2.a.k.1.2 ✓ 2 4.3 odd 2
1150.2.a.p.1.1 yes 2 20.19 odd 2
1150.2.b.h.599.1 4 20.7 even 4
1150.2.b.h.599.4 4 20.3 even 4
9200.2.a.bp.1.2 2 5.4 even 2
9200.2.a.bw.1.1 2 1.1 even 1 trivial