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Properties

Label 9800.2.a.db.1.4
Level $9800$
Weight $2$
Character 9800.1
Self dual yes
Analytic conductor $78.253$
Analytic rank $1$
Dimension $10$
Inner twists $2$

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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 = [10,0,0,0,0,0,0,0,14,0,-12,0,0,0,0,0,0,0,0,0,0,0,-16] 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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 22x^{8} - 16x^{7} + 146x^{6} + 200x^{5} - 206x^{4} - 440x^{3} - 124x^{2} + 72x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1960)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.50994\) 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)
 
\(f(q)\) \(=\) \(q-1.12212 q^{3} -1.74084 q^{9} -4.94252 q^{11} +4.64101 q^{13} -5.72505 q^{17} -7.53763 q^{19} +6.87122 q^{23} +5.31980 q^{27} +8.11636 q^{29} +3.87614 q^{31} +5.54611 q^{33} -5.18786 q^{37} -5.20778 q^{39} +9.45170 q^{41} +0.706904 q^{43} +2.20616 q^{47} +6.42420 q^{51} +1.32166 q^{53} +8.45814 q^{57} +3.94064 q^{59} +6.07542 q^{61} +8.24674 q^{67} -7.71035 q^{69} -4.50952 q^{71} +2.93470 q^{73} -11.0249 q^{79} -0.746943 q^{81} -5.27356 q^{83} -9.10755 q^{87} +9.49970 q^{89} -4.34950 q^{93} -1.74808 q^{97} +8.60415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 14 q^{9} - 12 q^{11} - 16 q^{23} + 12 q^{29} - 36 q^{37} - 20 q^{39} - 24 q^{43} - 36 q^{51} - 8 q^{53} - 16 q^{57} - 40 q^{67} - 8 q^{71} - 4 q^{79} + 50 q^{81} - 48 q^{93} - 72 q^{99}+O(q^{100}) \) 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
\(3\) −1.12212 −0.647857 −0.323929 − 0.946081i \(-0.605004\pi\)
−0.323929 + 0.946081i \(0.605004\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.74084 −0.580281
\(10\) 0 0
\(11\) −4.94252 −1.49023 −0.745113 − 0.666938i \(-0.767604\pi\)
−0.745113 + 0.666938i \(0.767604\pi\)
\(12\) 0 0
\(13\) 4.64101 1.28718 0.643592 − 0.765369i \(-0.277443\pi\)
0.643592 + 0.765369i \(0.277443\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.72505 −1.38853 −0.694264 − 0.719720i \(-0.744270\pi\)
−0.694264 + 0.719720i \(0.744270\pi\)
\(18\) 0 0
\(19\) −7.53763 −1.72925 −0.864625 − 0.502417i \(-0.832444\pi\)
−0.864625 + 0.502417i \(0.832444\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.87122 1.43275 0.716374 − 0.697716i \(-0.245800\pi\)
0.716374 + 0.697716i \(0.245800\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.31980 1.02380
\(28\) 0 0
\(29\) 8.11636 1.50717 0.753585 − 0.657350i \(-0.228323\pi\)
0.753585 + 0.657350i \(0.228323\pi\)
\(30\) 0 0
\(31\) 3.87614 0.696175 0.348087 − 0.937462i \(-0.386831\pi\)
0.348087 + 0.937462i \(0.386831\pi\)
\(32\) 0 0
\(33\) 5.54611 0.965454
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.18786 −0.852879 −0.426439 − 0.904516i \(-0.640232\pi\)
−0.426439 + 0.904516i \(0.640232\pi\)
\(38\) 0 0
\(39\) −5.20778 −0.833912
\(40\) 0 0
\(41\) 9.45170 1.47611 0.738054 − 0.674742i \(-0.235745\pi\)
0.738054 + 0.674742i \(0.235745\pi\)
\(42\) 0 0
\(43\) 0.706904 0.107802 0.0539009 − 0.998546i \(-0.482834\pi\)
0.0539009 + 0.998546i \(0.482834\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.20616 0.321802 0.160901 − 0.986971i \(-0.448560\pi\)
0.160901 + 0.986971i \(0.448560\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 6.42420 0.899568
\(52\) 0 0
\(53\) 1.32166 0.181544 0.0907721 − 0.995872i \(-0.471067\pi\)
0.0907721 + 0.995872i \(0.471067\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.45814 1.12031
\(58\) 0 0
\(59\) 3.94064 0.513027 0.256514 − 0.966541i \(-0.417426\pi\)
0.256514 + 0.966541i \(0.417426\pi\)
\(60\) 0 0
\(61\) 6.07542 0.777878 0.388939 − 0.921264i \(-0.372842\pi\)
0.388939 + 0.921264i \(0.372842\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.24674 1.00750 0.503750 − 0.863850i \(-0.331953\pi\)
0.503750 + 0.863850i \(0.331953\pi\)
\(68\) 0 0
\(69\) −7.71035 −0.928217
\(70\) 0 0
\(71\) −4.50952 −0.535182 −0.267591 − 0.963533i \(-0.586228\pi\)
−0.267591 + 0.963533i \(0.586228\pi\)
\(72\) 0 0
\(73\) 2.93470 0.343481 0.171741 − 0.985142i \(-0.445061\pi\)
0.171741 + 0.985142i \(0.445061\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −11.0249 −1.24040 −0.620201 − 0.784443i \(-0.712949\pi\)
−0.620201 + 0.784443i \(0.712949\pi\)
\(80\) 0 0
\(81\) −0.746943 −0.0829937
\(82\) 0 0
\(83\) −5.27356 −0.578848 −0.289424 − 0.957201i \(-0.593464\pi\)
−0.289424 + 0.957201i \(0.593464\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.10755 −0.976432
\(88\) 0 0
\(89\) 9.49970 1.00697 0.503483 − 0.864005i \(-0.332052\pi\)
0.503483 + 0.864005i \(0.332052\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.34950 −0.451022
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.74808 −0.177491 −0.0887454 − 0.996054i \(-0.528286\pi\)
−0.0887454 + 0.996054i \(0.528286\pi\)
\(98\) 0 0
\(99\) 8.60415 0.864749
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.db.1.4 10
5.2 odd 4 1960.2.g.g.1569.13 yes 20
5.3 odd 4 1960.2.g.g.1569.7 ✓ 20
5.4 even 2 9800.2.a.dc.1.7 10
7.6 odd 2 inner 9800.2.a.db.1.7 10
35.13 even 4 1960.2.g.g.1569.14 yes 20
35.27 even 4 1960.2.g.g.1569.8 yes 20
35.34 odd 2 9800.2.a.dc.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1960.2.g.g.1569.7 ✓ 20 5.3 odd 4
1960.2.g.g.1569.8 yes 20 35.27 even 4
1960.2.g.g.1569.13 yes 20 5.2 odd 4
1960.2.g.g.1569.14 yes 20 35.13 even 4
9800.2.a.db.1.4 10 1.1 even 1 trivial
9800.2.a.db.1.7 10 7.6 odd 2 inner
9800.2.a.dc.1.4 10 35.34 odd 2
9800.2.a.dc.1.7 10 5.4 even 2