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Properties

Label 9200.2.a.ch.1.1
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $3$
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 = [3,0,2,0,0,0,4,0,-3,0,0] 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: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 4600)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) 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)
 
\(f(q)\) \(=\) \(q-0.801938 q^{3} -1.60388 q^{7} -2.35690 q^{9} +6.09783 q^{11} +1.35690 q^{13} +1.60388 q^{17} -1.78017 q^{19} +1.28621 q^{21} -1.00000 q^{23} +4.29590 q^{27} -4.82908 q^{29} -6.15883 q^{31} -4.89008 q^{33} +9.87800 q^{37} -1.08815 q^{39} -8.89977 q^{41} -11.3840 q^{43} +9.85086 q^{47} -4.42758 q^{49} -1.28621 q^{51} -9.20775 q^{53} +1.42758 q^{57} +7.27413 q^{59} +0.933624 q^{61} +3.78017 q^{63} +9.42758 q^{67} +0.801938 q^{69} -12.6189 q^{71} +2.60925 q^{73} -9.78017 q^{77} +5.16421 q^{79} +3.62565 q^{81} +15.4034 q^{83} +3.87263 q^{87} -15.7017 q^{89} -2.17629 q^{91} +4.93900 q^{93} -12.8901 q^{97} -14.3720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 4 q^{7} - 3 q^{9} - 4 q^{17} - 4 q^{19} + 12 q^{21} - 3 q^{23} - q^{27} - 4 q^{29} - 10 q^{31} - 14 q^{33} + 10 q^{37} - 7 q^{39} - 4 q^{41} - 24 q^{43} + 16 q^{47} + 3 q^{49} - 12 q^{51}+ \cdots - 14 q^{99}+O(q^{100}) \) 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
\(3\) −0.801938 −0.462999 −0.231499 − 0.972835i \(-0.574363\pi\)
−0.231499 + 0.972835i \(0.574363\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.60388 −0.606208 −0.303104 − 0.952957i \(-0.598023\pi\)
−0.303104 + 0.952957i \(0.598023\pi\)
\(8\) 0 0
\(9\) −2.35690 −0.785632
\(10\) 0 0
\(11\) 6.09783 1.83857 0.919283 − 0.393597i \(-0.128769\pi\)
0.919283 + 0.393597i \(0.128769\pi\)
\(12\) 0 0
\(13\) 1.35690 0.376335 0.188168 − 0.982137i \(-0.439745\pi\)
0.188168 + 0.982137i \(0.439745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.60388 0.388997 0.194498 − 0.980903i \(-0.437692\pi\)
0.194498 + 0.980903i \(0.437692\pi\)
\(18\) 0 0
\(19\) −1.78017 −0.408398 −0.204199 − 0.978929i \(-0.565459\pi\)
−0.204199 + 0.978929i \(0.565459\pi\)
\(20\) 0 0
\(21\) 1.28621 0.280674
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.29590 0.826746
\(28\) 0 0
\(29\) −4.82908 −0.896739 −0.448369 − 0.893848i \(-0.647995\pi\)
−0.448369 + 0.893848i \(0.647995\pi\)
\(30\) 0 0
\(31\) −6.15883 −1.10616 −0.553080 − 0.833128i \(-0.686547\pi\)
−0.553080 + 0.833128i \(0.686547\pi\)
\(32\) 0 0
\(33\) −4.89008 −0.851254
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.87800 1.62393 0.811967 − 0.583704i \(-0.198397\pi\)
0.811967 + 0.583704i \(0.198397\pi\)
\(38\) 0 0
\(39\) −1.08815 −0.174243
\(40\) 0 0
\(41\) −8.89977 −1.38991 −0.694955 − 0.719053i \(-0.744576\pi\)
−0.694955 + 0.719053i \(0.744576\pi\)
\(42\) 0 0
\(43\) −11.3840 −1.73605 −0.868025 − 0.496520i \(-0.834611\pi\)
−0.868025 + 0.496520i \(0.834611\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.85086 1.43689 0.718447 − 0.695581i \(-0.244853\pi\)
0.718447 + 0.695581i \(0.244853\pi\)
\(48\) 0 0
\(49\) −4.42758 −0.632512
\(50\) 0 0
\(51\) −1.28621 −0.180105
\(52\) 0 0
\(53\) −9.20775 −1.26478 −0.632391 − 0.774649i \(-0.717926\pi\)
−0.632391 + 0.774649i \(0.717926\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.42758 0.189088
\(58\) 0 0
\(59\) 7.27413 0.947011 0.473505 − 0.880791i \(-0.342988\pi\)
0.473505 + 0.880791i \(0.342988\pi\)
\(60\) 0 0
\(61\) 0.933624 0.119538 0.0597692 − 0.998212i \(-0.480964\pi\)
0.0597692 + 0.998212i \(0.480964\pi\)
\(62\) 0 0
\(63\) 3.78017 0.476256
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.42758 1.15176 0.575881 − 0.817533i \(-0.304659\pi\)
0.575881 + 0.817533i \(0.304659\pi\)
\(68\) 0 0
\(69\) 0.801938 0.0965420
\(70\) 0 0
\(71\) −12.6189 −1.49759 −0.748796 − 0.662800i \(-0.769368\pi\)
−0.748796 + 0.662800i \(0.769368\pi\)
\(72\) 0 0
\(73\) 2.60925 0.305390 0.152695 − 0.988273i \(-0.451205\pi\)
0.152695 + 0.988273i \(0.451205\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −9.78017 −1.11455
\(78\) 0 0
\(79\) 5.16421 0.581019 0.290510 − 0.956872i \(-0.406175\pi\)
0.290510 + 0.956872i \(0.406175\pi\)
\(80\) 0 0
\(81\) 3.62565 0.402850
\(82\) 0 0
\(83\) 15.4034 1.69075 0.845373 − 0.534177i \(-0.179379\pi\)
0.845373 + 0.534177i \(0.179379\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.87263 0.415189
\(88\) 0 0
\(89\) −15.7017 −1.66438 −0.832189 − 0.554492i \(-0.812913\pi\)
−0.832189 + 0.554492i \(0.812913\pi\)
\(90\) 0 0
\(91\) −2.17629 −0.228137
\(92\) 0 0
\(93\) 4.93900 0.512151
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.8901 −1.30879 −0.654395 − 0.756153i \(-0.727077\pi\)
−0.654395 + 0.756153i \(0.727077\pi\)
\(98\) 0 0
\(99\) −14.3720 −1.44444
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.ch.1.1 3
4.3 odd 2 4600.2.a.w.1.3 ✓ 3
5.4 even 2 9200.2.a.cb.1.3 3
20.3 even 4 4600.2.e.s.4049.5 6
20.7 even 4 4600.2.e.s.4049.2 6
20.19 odd 2 4600.2.a.z.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.w.1.3 ✓ 3 4.3 odd 2
4600.2.a.z.1.1 yes 3 20.19 odd 2
4600.2.e.s.4049.2 6 20.7 even 4
4600.2.e.s.4049.5 6 20.3 even 4
9200.2.a.cb.1.3 3 5.4 even 2
9200.2.a.ch.1.1 3 1.1 even 1 trivial