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Properties

Label 9675.2.a.da.1.3
Level $9675$
Weight $2$
Character 9675.1
Self dual yes
Analytic conductor $77.255$
Analytic rank $1$
Dimension $20$
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] = [9675,2,Mod(1,9675)] 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("9675.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9675 = 3^{2} \cdot 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9675.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,16,0,0,-8,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(77.2552639556\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 28 x^{18} + 326 x^{16} - 2052 x^{14} + 7613 x^{12} - 17056 x^{10} + 22796 x^{8} - 17428 x^{6} + \cdots + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1935)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.19006\) of defining polynomial
Character \(\chi\) \(=\) 9675.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.19006 q^{2} +2.79638 q^{4} -4.99426 q^{7} -1.74413 q^{8} -5.42090 q^{11} -5.08752 q^{13} +10.9377 q^{14} -1.77300 q^{16} -2.09728 q^{17} +7.09005 q^{19} +11.8721 q^{22} -1.86354 q^{23} +11.1420 q^{26} -13.9659 q^{28} -1.21100 q^{29} -3.60902 q^{31} +7.37126 q^{32} +4.59317 q^{34} +0.533626 q^{37} -15.5277 q^{38} +8.69797 q^{41} +1.00000 q^{43} -15.1589 q^{44} +4.08128 q^{46} +1.27299 q^{47} +17.9426 q^{49} -14.2267 q^{52} +6.46214 q^{53} +8.71064 q^{56} +2.65217 q^{58} +11.2530 q^{59} -11.8390 q^{61} +7.90398 q^{62} -12.5975 q^{64} -12.1852 q^{67} -5.86479 q^{68} +6.49765 q^{71} +0.220835 q^{73} -1.16868 q^{74} +19.8265 q^{76} +27.0733 q^{77} +4.51633 q^{79} -19.0491 q^{82} +9.92115 q^{83} -2.19006 q^{86} +9.45477 q^{88} -11.1048 q^{89} +25.4084 q^{91} -5.21118 q^{92} -2.78794 q^{94} +9.19491 q^{97} -39.2954 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{4} - 8 q^{7} - 32 q^{13} + 8 q^{16} - 16 q^{22} - 24 q^{28} - 12 q^{31} + 24 q^{34} - 52 q^{37} + 20 q^{43} + 8 q^{46} + 32 q^{49} - 92 q^{52} - 28 q^{58} + 12 q^{61} - 8 q^{64} - 40 q^{67}+ \cdots - 76 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.19006 −1.54861 −0.774305 − 0.632813i \(-0.781900\pi\)
−0.774305 + 0.632813i \(0.781900\pi\)
\(3\) 0 0
\(4\) 2.79638 1.39819
\(5\) 0 0
\(6\) 0 0
\(7\) −4.99426 −1.88765 −0.943826 − 0.330444i \(-0.892801\pi\)
−0.943826 + 0.330444i \(0.892801\pi\)
\(8\) −1.74413 −0.616644
\(9\) 0 0
\(10\) 0 0
\(11\) −5.42090 −1.63446 −0.817231 − 0.576310i \(-0.804492\pi\)
−0.817231 + 0.576310i \(0.804492\pi\)
\(12\) 0 0
\(13\) −5.08752 −1.41102 −0.705512 − 0.708698i \(-0.749283\pi\)
−0.705512 + 0.708698i \(0.749283\pi\)
\(14\) 10.9377 2.92323
\(15\) 0 0
\(16\) −1.77300 −0.443251
\(17\) −2.09728 −0.508664 −0.254332 − 0.967117i \(-0.581856\pi\)
−0.254332 + 0.967117i \(0.581856\pi\)
\(18\) 0 0
\(19\) 7.09005 1.62657 0.813284 − 0.581866i \(-0.197677\pi\)
0.813284 + 0.581866i \(0.197677\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 11.8721 2.53114
\(23\) −1.86354 −0.388575 −0.194288 − 0.980945i \(-0.562240\pi\)
−0.194288 + 0.980945i \(0.562240\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 11.1420 2.18513
\(27\) 0 0
\(28\) −13.9659 −2.63930
\(29\) −1.21100 −0.224878 −0.112439 − 0.993659i \(-0.535866\pi\)
−0.112439 + 0.993659i \(0.535866\pi\)
\(30\) 0 0
\(31\) −3.60902 −0.648199 −0.324099 − 0.946023i \(-0.605061\pi\)
−0.324099 + 0.946023i \(0.605061\pi\)
\(32\) 7.37126 1.30307
\(33\) 0 0
\(34\) 4.59317 0.787722
\(35\) 0 0
\(36\) 0 0
\(37\) 0.533626 0.0877276 0.0438638 − 0.999038i \(-0.486033\pi\)
0.0438638 + 0.999038i \(0.486033\pi\)
\(38\) −15.5277 −2.51892
\(39\) 0 0
\(40\) 0 0
\(41\) 8.69797 1.35839 0.679197 − 0.733956i \(-0.262328\pi\)
0.679197 + 0.733956i \(0.262328\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499
\(44\) −15.1589 −2.28529
\(45\) 0 0
\(46\) 4.08128 0.601752
\(47\) 1.27299 0.185685 0.0928425 − 0.995681i \(-0.470405\pi\)
0.0928425 + 0.995681i \(0.470405\pi\)
\(48\) 0 0
\(49\) 17.9426 2.56323
\(50\) 0 0
\(51\) 0 0
\(52\) −14.2267 −1.97288
\(53\) 6.46214 0.887643 0.443822 − 0.896115i \(-0.353622\pi\)
0.443822 + 0.896115i \(0.353622\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 8.71064 1.16401
\(57\) 0 0
\(58\) 2.65217 0.348248
\(59\) 11.2530 1.46501 0.732507 − 0.680759i \(-0.238350\pi\)
0.732507 + 0.680759i \(0.238350\pi\)
\(60\) 0 0
\(61\) −11.8390 −1.51583 −0.757916 − 0.652352i \(-0.773782\pi\)
−0.757916 + 0.652352i \(0.773782\pi\)
\(62\) 7.90398 1.00381
\(63\) 0 0
\(64\) −12.5975 −1.57469
\(65\) 0 0
\(66\) 0 0
\(67\) −12.1852 −1.48865 −0.744327 − 0.667815i \(-0.767230\pi\)
−0.744327 + 0.667815i \(0.767230\pi\)
\(68\) −5.86479 −0.711210
\(69\) 0 0
\(70\) 0 0
\(71\) 6.49765 0.771129 0.385564 − 0.922681i \(-0.374007\pi\)
0.385564 + 0.922681i \(0.374007\pi\)
\(72\) 0 0
\(73\) 0.220835 0.0258468 0.0129234 − 0.999916i \(-0.495886\pi\)
0.0129234 + 0.999916i \(0.495886\pi\)
\(74\) −1.16868 −0.135856
\(75\) 0 0
\(76\) 19.8265 2.27426
\(77\) 27.0733 3.08529
\(78\) 0 0
\(79\) 4.51633 0.508127 0.254063 − 0.967188i \(-0.418233\pi\)
0.254063 + 0.967188i \(0.418233\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −19.0491 −2.10362
\(83\) 9.92115 1.08899 0.544494 − 0.838765i \(-0.316722\pi\)
0.544494 + 0.838765i \(0.316722\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.19006 −0.236161
\(87\) 0 0
\(88\) 9.45477 1.00788
\(89\) −11.1048 −1.17711 −0.588554 − 0.808458i \(-0.700303\pi\)
−0.588554 + 0.808458i \(0.700303\pi\)
\(90\) 0 0
\(91\) 25.4084 2.66352
\(92\) −5.21118 −0.543303
\(93\) 0 0
\(94\) −2.78794 −0.287554
\(95\) 0 0
\(96\) 0 0
\(97\) 9.19491 0.933602 0.466801 − 0.884362i \(-0.345407\pi\)
0.466801 + 0.884362i \(0.345407\pi\)
\(98\) −39.2954 −3.96944
\(99\) 0 0
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 9675.2.a.da.1.3 20
3.2 odd 2 inner 9675.2.a.da.1.18 20
5.2 odd 4 1935.2.b.g.1549.6 yes 40
5.3 odd 4 1935.2.b.g.1549.36 yes 40
5.4 even 2 9675.2.a.db.1.18 20
15.2 even 4 1935.2.b.g.1549.35 yes 40
15.8 even 4 1935.2.b.g.1549.5 ✓ 40
15.14 odd 2 9675.2.a.db.1.3 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1935.2.b.g.1549.5 ✓ 40 15.8 even 4
1935.2.b.g.1549.6 yes 40 5.2 odd 4
1935.2.b.g.1549.35 yes 40 15.2 even 4
1935.2.b.g.1549.36 yes 40 5.3 odd 4
9675.2.a.da.1.3 20 1.1 even 1 trivial
9675.2.a.da.1.18 20 3.2 odd 2 inner
9675.2.a.db.1.3 20 15.14 odd 2
9675.2.a.db.1.18 20 5.4 even 2