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Properties

Label 9675.2.a.cs.1.7
Level $9675$
Weight $2$
Character 9675.1
Self dual yes
Analytic conductor $77.255$
Analytic rank $0$
Dimension $9$
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] = [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 = [9,-3,0,13,0,0,8,-6,0,0,-1,0,3] 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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 11x^{7} + 36x^{6} + 29x^{5} - 120x^{4} - 13x^{3} + 127x^{2} - 4x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3225)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.07358\) 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)
 
\(f(q)\) \(=\) \(q+1.07358 q^{2} -0.847434 q^{4} -4.32861 q^{7} -3.05694 q^{8} +1.99553 q^{11} -2.07300 q^{13} -4.64709 q^{14} -1.58699 q^{16} -0.350661 q^{17} -7.90746 q^{19} +2.14235 q^{22} -6.87707 q^{23} -2.22552 q^{26} +3.66821 q^{28} -1.51061 q^{29} -3.13471 q^{31} +4.41012 q^{32} -0.376462 q^{34} -6.38218 q^{37} -8.48926 q^{38} +2.93661 q^{41} -1.00000 q^{43} -1.69108 q^{44} -7.38306 q^{46} +0.501373 q^{47} +11.7369 q^{49} +1.75673 q^{52} -4.74935 q^{53} +13.2323 q^{56} -1.62175 q^{58} -7.26909 q^{59} +2.61746 q^{61} -3.36535 q^{62} +7.90858 q^{64} +14.8131 q^{67} +0.297162 q^{68} +3.40434 q^{71} -5.12823 q^{73} -6.85175 q^{74} +6.70105 q^{76} -8.63785 q^{77} -10.5631 q^{79} +3.15268 q^{82} -10.5696 q^{83} -1.07358 q^{86} -6.10020 q^{88} +16.1009 q^{89} +8.97320 q^{91} +5.82786 q^{92} +0.538262 q^{94} -4.65451 q^{97} +12.6004 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} + 13 q^{4} + 8 q^{7} - 6 q^{8} - q^{11} + 3 q^{13} - 3 q^{14} + 21 q^{16} - 11 q^{17} + 7 q^{19} + 15 q^{22} - 30 q^{23} + 22 q^{26} + 32 q^{28} + 6 q^{29} + 13 q^{31} - 4 q^{34} + 5 q^{37}+ \cdots + 32 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.07358 0.759133 0.379567 − 0.925164i \(-0.376073\pi\)
0.379567 + 0.925164i \(0.376073\pi\)
\(3\) 0 0
\(4\) −0.847434 −0.423717
\(5\) 0 0
\(6\) 0 0
\(7\) −4.32861 −1.63606 −0.818030 − 0.575175i \(-0.804934\pi\)
−0.818030 + 0.575175i \(0.804934\pi\)
\(8\) −3.05694 −1.08079
\(9\) 0 0
\(10\) 0 0
\(11\) 1.99553 0.601674 0.300837 − 0.953676i \(-0.402734\pi\)
0.300837 + 0.953676i \(0.402734\pi\)
\(12\) 0 0
\(13\) −2.07300 −0.574947 −0.287473 − 0.957789i \(-0.592815\pi\)
−0.287473 + 0.957789i \(0.592815\pi\)
\(14\) −4.64709 −1.24199
\(15\) 0 0
\(16\) −1.58699 −0.396747
\(17\) −0.350661 −0.0850478 −0.0425239 − 0.999095i \(-0.513540\pi\)
−0.0425239 + 0.999095i \(0.513540\pi\)
\(18\) 0 0
\(19\) −7.90746 −1.81409 −0.907047 − 0.421029i \(-0.861669\pi\)
−0.907047 + 0.421029i \(0.861669\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.14235 0.456750
\(23\) −6.87707 −1.43397 −0.716984 − 0.697090i \(-0.754478\pi\)
−0.716984 + 0.697090i \(0.754478\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.22552 −0.436461
\(27\) 0 0
\(28\) 3.66821 0.693227
\(29\) −1.51061 −0.280513 −0.140256 − 0.990115i \(-0.544793\pi\)
−0.140256 + 0.990115i \(0.544793\pi\)
\(30\) 0 0
\(31\) −3.13471 −0.563010 −0.281505 − 0.959560i \(-0.590834\pi\)
−0.281505 + 0.959560i \(0.590834\pi\)
\(32\) 4.41012 0.779607
\(33\) 0 0
\(34\) −0.376462 −0.0645626
\(35\) 0 0
\(36\) 0 0
\(37\) −6.38218 −1.04922 −0.524612 − 0.851342i \(-0.675789\pi\)
−0.524612 + 0.851342i \(0.675789\pi\)
\(38\) −8.48926 −1.37714
\(39\) 0 0
\(40\) 0 0
\(41\) 2.93661 0.458622 0.229311 − 0.973353i \(-0.426353\pi\)
0.229311 + 0.973353i \(0.426353\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499
\(44\) −1.69108 −0.254939
\(45\) 0 0
\(46\) −7.38306 −1.08857
\(47\) 0.501373 0.0731327 0.0365664 − 0.999331i \(-0.488358\pi\)
0.0365664 + 0.999331i \(0.488358\pi\)
\(48\) 0 0
\(49\) 11.7369 1.67669
\(50\) 0 0
\(51\) 0 0
\(52\) 1.75673 0.243615
\(53\) −4.74935 −0.652373 −0.326187 − 0.945305i \(-0.605764\pi\)
−0.326187 + 0.945305i \(0.605764\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 13.2323 1.76824
\(57\) 0 0
\(58\) −1.62175 −0.212946
\(59\) −7.26909 −0.946354 −0.473177 − 0.880967i \(-0.656893\pi\)
−0.473177 + 0.880967i \(0.656893\pi\)
\(60\) 0 0
\(61\) 2.61746 0.335132 0.167566 − 0.985861i \(-0.446409\pi\)
0.167566 + 0.985861i \(0.446409\pi\)
\(62\) −3.36535 −0.427400
\(63\) 0 0
\(64\) 7.90858 0.988572
\(65\) 0 0
\(66\) 0 0
\(67\) 14.8131 1.80970 0.904852 − 0.425727i \(-0.139982\pi\)
0.904852 + 0.425727i \(0.139982\pi\)
\(68\) 0.297162 0.0360362
\(69\) 0 0
\(70\) 0 0
\(71\) 3.40434 0.404021 0.202010 − 0.979383i \(-0.435252\pi\)
0.202010 + 0.979383i \(0.435252\pi\)
\(72\) 0 0
\(73\) −5.12823 −0.600214 −0.300107 − 0.953905i \(-0.597022\pi\)
−0.300107 + 0.953905i \(0.597022\pi\)
\(74\) −6.85175 −0.796500
\(75\) 0 0
\(76\) 6.70105 0.768663
\(77\) −8.63785 −0.984374
\(78\) 0 0
\(79\) −10.5631 −1.18844 −0.594221 − 0.804302i \(-0.702539\pi\)
−0.594221 + 0.804302i \(0.702539\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.15268 0.348155
\(83\) −10.5696 −1.16016 −0.580082 − 0.814558i \(-0.696979\pi\)
−0.580082 + 0.814558i \(0.696979\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.07358 −0.115767
\(87\) 0 0
\(88\) −6.10020 −0.650283
\(89\) 16.1009 1.70669 0.853344 − 0.521348i \(-0.174571\pi\)
0.853344 + 0.521348i \(0.174571\pi\)
\(90\) 0 0
\(91\) 8.97320 0.940647
\(92\) 5.82786 0.607597
\(93\) 0 0
\(94\) 0.538262 0.0555175
\(95\) 0 0
\(96\) 0 0
\(97\) −4.65451 −0.472594 −0.236297 − 0.971681i \(-0.575934\pi\)
−0.236297 + 0.971681i \(0.575934\pi\)
\(98\) 12.6004 1.27283
\(99\) 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.cs.1.7 9
3.2 odd 2 3225.2.a.bf.1.3 yes 9
5.4 even 2 9675.2.a.ct.1.3 9
15.14 odd 2 3225.2.a.be.1.7 ✓ 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3225.2.a.be.1.7 ✓ 9 15.14 odd 2
3225.2.a.bf.1.3 yes 9 3.2 odd 2
9675.2.a.cs.1.7 9 1.1 even 1 trivial
9675.2.a.ct.1.3 9 5.4 even 2