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Properties

Label 9800.2.a.cf.1.3
Level $9800$
Weight $2$
Character 9800.1
Self dual yes
Analytic conductor $78.253$
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] = [9800,2,Mod(1,9800)] 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("9800.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,0,0,0,0,9,0,3,0,3,0,0,0,6,0,-3,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1944.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 9x - 6 \) 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 280)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.58423\) of defining polynomial
Character \(\chi\) \(=\) 9800.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.58423 q^{3} +3.67822 q^{9} +1.67822 q^{11} -4.84667 q^{13} +2.00000 q^{17} -6.84667 q^{19} +2.26245 q^{23} +1.75268 q^{27} +3.32178 q^{29} -9.16845 q^{31} +4.33690 q^{33} +2.84667 q^{37} -12.5249 q^{39} -9.52489 q^{41} -6.58423 q^{43} -12.2031 q^{47} +5.16845 q^{51} -7.49023 q^{53} -17.6933 q^{57} -8.00000 q^{59} +6.49023 q^{61} +5.75268 q^{67} +5.84667 q^{69} -11.6933 q^{73} +5.69334 q^{79} -6.50535 q^{81} -12.5842 q^{83} +8.58423 q^{87} +5.84667 q^{89} -23.6933 q^{93} -2.00000 q^{97} +6.17287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 9 q^{9} + 3 q^{11} + 3 q^{13} + 6 q^{17} - 3 q^{19} - 3 q^{23} - 18 q^{27} + 12 q^{29} - 12 q^{31} - 18 q^{33} - 9 q^{37} - 18 q^{39} - 9 q^{41} - 12 q^{43} - 15 q^{47} - 9 q^{53} - 18 q^{57} - 24 q^{59}+ \cdots + 63 q^{99}+O(q^{100}) \) 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
\(3\) 2.58423 1.49200 0.746002 − 0.665944i \(-0.231971\pi\)
0.746002 + 0.665944i \(0.231971\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 3.67822 1.22607
\(10\) 0 0
\(11\) 1.67822 0.506003 0.253001 − 0.967466i \(-0.418582\pi\)
0.253001 + 0.967466i \(0.418582\pi\)
\(12\) 0 0
\(13\) −4.84667 −1.34422 −0.672112 − 0.740449i \(-0.734613\pi\)
−0.672112 + 0.740449i \(0.734613\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −6.84667 −1.57073 −0.785367 − 0.619030i \(-0.787526\pi\)
−0.785367 + 0.619030i \(0.787526\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.26245 0.471753 0.235876 − 0.971783i \(-0.424204\pi\)
0.235876 + 0.971783i \(0.424204\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.75268 0.337303
\(28\) 0 0
\(29\) 3.32178 0.616839 0.308419 − 0.951250i \(-0.400200\pi\)
0.308419 + 0.951250i \(0.400200\pi\)
\(30\) 0 0
\(31\) −9.16845 −1.64670 −0.823351 − 0.567532i \(-0.807898\pi\)
−0.823351 + 0.567532i \(0.807898\pi\)
\(32\) 0 0
\(33\) 4.33690 0.754958
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.84667 0.467990 0.233995 − 0.972238i \(-0.424820\pi\)
0.233995 + 0.972238i \(0.424820\pi\)
\(38\) 0 0
\(39\) −12.5249 −2.00559
\(40\) 0 0
\(41\) −9.52489 −1.48754 −0.743769 − 0.668437i \(-0.766964\pi\)
−0.743769 + 0.668437i \(0.766964\pi\)
\(42\) 0 0
\(43\) −6.58423 −1.00408 −0.502042 − 0.864843i \(-0.667418\pi\)
−0.502042 + 0.864843i \(0.667418\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.2031 −1.78001 −0.890004 − 0.455954i \(-0.849298\pi\)
−0.890004 + 0.455954i \(0.849298\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 5.16845 0.723728
\(52\) 0 0
\(53\) −7.49023 −1.02886 −0.514431 − 0.857532i \(-0.671997\pi\)
−0.514431 + 0.857532i \(0.671997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −17.6933 −2.34354
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 6.49023 0.830989 0.415494 − 0.909596i \(-0.363609\pi\)
0.415494 + 0.909596i \(0.363609\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.75268 0.702801 0.351401 − 0.936225i \(-0.385706\pi\)
0.351401 + 0.936225i \(0.385706\pi\)
\(68\) 0 0
\(69\) 5.84667 0.703857
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −11.6933 −1.36860 −0.684301 − 0.729199i \(-0.739893\pi\)
−0.684301 + 0.729199i \(0.739893\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.69334 0.640551 0.320276 − 0.947324i \(-0.396224\pi\)
0.320276 + 0.947324i \(0.396224\pi\)
\(80\) 0 0
\(81\) −6.50535 −0.722817
\(82\) 0 0
\(83\) −12.5842 −1.38130 −0.690649 − 0.723190i \(-0.742675\pi\)
−0.690649 + 0.723190i \(0.742675\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.58423 0.920326
\(88\) 0 0
\(89\) 5.84667 0.619746 0.309873 − 0.950778i \(-0.399713\pi\)
0.309873 + 0.950778i \(0.399713\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −23.6933 −2.45689
\(94\) 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
\(99\) 6.17287 0.620397
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 9800.2.a.cf.1.3 3
5.4 even 2 1960.2.a.v.1.1 3
7.3 odd 6 1400.2.q.j.401.3 6
7.5 odd 6 1400.2.q.j.1201.3 6
7.6 odd 2 9800.2.a.ce.1.1 3
20.19 odd 2 3920.2.a.cb.1.3 3
35.3 even 12 1400.2.bh.i.849.2 12
35.4 even 6 1960.2.q.w.961.3 6
35.9 even 6 1960.2.q.w.361.3 6
35.12 even 12 1400.2.bh.i.249.2 12
35.17 even 12 1400.2.bh.i.849.5 12
35.19 odd 6 280.2.q.e.81.1 ✓ 6
35.24 odd 6 280.2.q.e.121.1 yes 6
35.33 even 12 1400.2.bh.i.249.5 12
35.34 odd 2 1960.2.a.w.1.3 3
105.59 even 6 2520.2.bi.q.1801.1 6
105.89 even 6 2520.2.bi.q.361.1 6
140.19 even 6 560.2.q.l.81.3 6
140.59 even 6 560.2.q.l.401.3 6
140.139 even 2 3920.2.a.cc.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.e.81.1 ✓ 6 35.19 odd 6
280.2.q.e.121.1 yes 6 35.24 odd 6
560.2.q.l.81.3 6 140.19 even 6
560.2.q.l.401.3 6 140.59 even 6
1400.2.q.j.401.3 6 7.3 odd 6
1400.2.q.j.1201.3 6 7.5 odd 6
1400.2.bh.i.249.2 12 35.12 even 12
1400.2.bh.i.249.5 12 35.33 even 12
1400.2.bh.i.849.2 12 35.3 even 12
1400.2.bh.i.849.5 12 35.17 even 12
1960.2.a.v.1.1 3 5.4 even 2
1960.2.a.w.1.3 3 35.34 odd 2
1960.2.q.w.361.3 6 35.9 even 6
1960.2.q.w.961.3 6 35.4 even 6
2520.2.bi.q.361.1 6 105.89 even 6
2520.2.bi.q.1801.1 6 105.59 even 6
3920.2.a.cb.1.3 3 20.19 odd 2
3920.2.a.cc.1.1 3 140.139 even 2
9800.2.a.ce.1.1 3 7.6 odd 2
9800.2.a.cf.1.3 3 1.1 even 1 trivial