↑

Properties

Label 9675.2.a.da.1.4
Level $9675$
Weight $2$
Character 9675.1
Self dual yes
Analytic conductor $77.255$
Analytic rank $1$
Dimension $20$
Inner twists $2$

Related objects

Downloads

Learn more

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 1.4
Root \(-1.83296\) 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)
 
\(f(q)\) \(=\) \(q-1.83296 q^{2} +1.35974 q^{4} +3.64733 q^{7} +1.17356 q^{8} -0.124282 q^{11} -2.43266 q^{13} -6.68540 q^{14} -4.87058 q^{16} -2.46992 q^{17} -6.77389 q^{19} +0.227804 q^{22} -4.21466 q^{23} +4.45897 q^{26} +4.95943 q^{28} +9.62369 q^{29} -10.6152 q^{31} +6.58046 q^{32} +4.52726 q^{34} +6.85378 q^{37} +12.4163 q^{38} +11.0350 q^{41} +1.00000 q^{43} -0.168992 q^{44} +7.72530 q^{46} +1.18498 q^{47} +6.30298 q^{49} -3.30780 q^{52} -1.82984 q^{53} +4.28036 q^{56} -17.6399 q^{58} +0.0841352 q^{59} +0.606586 q^{61} +19.4573 q^{62} -2.32056 q^{64} +8.00020 q^{67} -3.35845 q^{68} +11.2901 q^{71} -16.6712 q^{73} -12.5627 q^{74} -9.21076 q^{76} -0.453297 q^{77} +10.0899 q^{79} -20.2268 q^{82} +4.80853 q^{83} -1.83296 q^{86} -0.145853 q^{88} -4.53049 q^{89} -8.87270 q^{91} -5.73086 q^{92} -2.17202 q^{94} -10.7693 q^{97} -11.5531 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{4} - 8 q^{7} - 32 q^{13} + 8 q^{16} - 16 q^{22} - 24 q^{28} - 12 q^{31} + 24 q^{34} - 52 q^{37} + 20 q^{43} + 8 q^{46} + 32 q^{49} - 92 q^{52} - 28 q^{58} + 12 q^{61} - 8 q^{64} - 40 q^{67}+ \cdots - 76 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.83296 −1.29610 −0.648049 − 0.761598i \(-0.724415\pi\)
−0.648049 + 0.761598i \(0.724415\pi\)
\(3\) 0 0
\(4\) 1.35974 0.679872
\(5\) 0 0
\(6\) 0 0
\(7\) 3.64733 1.37856 0.689280 − 0.724495i \(-0.257927\pi\)
0.689280 + 0.724495i \(0.257927\pi\)
\(8\) 1.17356 0.414917
\(9\) 0 0
\(10\) 0 0
\(11\) −0.124282 −0.0374725 −0.0187362 − 0.999824i \(-0.505964\pi\)
−0.0187362 + 0.999824i \(0.505964\pi\)
\(12\) 0 0
\(13\) −2.43266 −0.674698 −0.337349 − 0.941380i \(-0.609530\pi\)
−0.337349 + 0.941380i \(0.609530\pi\)
\(14\) −6.68540 −1.78675
\(15\) 0 0
\(16\) −4.87058 −1.21765
\(17\) −2.46992 −0.599042 −0.299521 − 0.954090i \(-0.596827\pi\)
−0.299521 + 0.954090i \(0.596827\pi\)
\(18\) 0 0
\(19\) −6.77389 −1.55404 −0.777018 − 0.629478i \(-0.783269\pi\)
−0.777018 + 0.629478i \(0.783269\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.227804 0.0485680
\(23\) −4.21466 −0.878817 −0.439408 − 0.898287i \(-0.644812\pi\)
−0.439408 + 0.898287i \(0.644812\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.45897 0.874476
\(27\) 0 0
\(28\) 4.95943 0.937245
\(29\) 9.62369 1.78708 0.893538 − 0.448988i \(-0.148216\pi\)
0.893538 + 0.448988i \(0.148216\pi\)
\(30\) 0 0
\(31\) −10.6152 −1.90655 −0.953274 − 0.302106i \(-0.902310\pi\)
−0.953274 + 0.302106i \(0.902310\pi\)
\(32\) 6.58046 1.16327
\(33\) 0 0
\(34\) 4.52726 0.776418
\(35\) 0 0
\(36\) 0 0
\(37\) 6.85378 1.12675 0.563377 − 0.826200i \(-0.309502\pi\)
0.563377 + 0.826200i \(0.309502\pi\)
\(38\) 12.4163 2.01418
\(39\) 0 0
\(40\) 0 0
\(41\) 11.0350 1.72338 0.861692 − 0.507432i \(-0.169405\pi\)
0.861692 + 0.507432i \(0.169405\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499
\(44\) −0.168992 −0.0254765
\(45\) 0 0
\(46\) 7.72530 1.13903
\(47\) 1.18498 0.172847 0.0864235 − 0.996258i \(-0.472456\pi\)
0.0864235 + 0.996258i \(0.472456\pi\)
\(48\) 0 0
\(49\) 6.30298 0.900426
\(50\) 0 0
\(51\) 0 0
\(52\) −3.30780 −0.458709
\(53\) −1.82984 −0.251348 −0.125674 − 0.992072i \(-0.540109\pi\)
−0.125674 + 0.992072i \(0.540109\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.28036 0.571988
\(57\) 0 0
\(58\) −17.6399 −2.31623
\(59\) 0.0841352 0.0109535 0.00547674 − 0.999985i \(-0.498257\pi\)
0.00547674 + 0.999985i \(0.498257\pi\)
\(60\) 0 0
\(61\) 0.606586 0.0776654 0.0388327 − 0.999246i \(-0.487636\pi\)
0.0388327 + 0.999246i \(0.487636\pi\)
\(62\) 19.4573 2.47108
\(63\) 0 0
\(64\) −2.32056 −0.290070
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00020 0.977380 0.488690 − 0.872458i \(-0.337475\pi\)
0.488690 + 0.872458i \(0.337475\pi\)
\(68\) −3.35845 −0.407272
\(69\) 0 0
\(70\) 0 0
\(71\) 11.2901 1.33989 0.669946 − 0.742410i \(-0.266317\pi\)
0.669946 + 0.742410i \(0.266317\pi\)
\(72\) 0 0
\(73\) −16.6712 −1.95122 −0.975609 − 0.219517i \(-0.929552\pi\)
−0.975609 + 0.219517i \(0.929552\pi\)
\(74\) −12.5627 −1.46039
\(75\) 0 0
\(76\) −9.21076 −1.05655
\(77\) −0.453297 −0.0516580
\(78\) 0 0
\(79\) 10.0899 1.13521 0.567604 − 0.823302i \(-0.307871\pi\)
0.567604 + 0.823302i \(0.307871\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −20.2268 −2.23368
\(83\) 4.80853 0.527804 0.263902 − 0.964549i \(-0.414990\pi\)
0.263902 + 0.964549i \(0.414990\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.83296 −0.197653
\(87\) 0 0
\(88\) −0.145853 −0.0155480
\(89\) −4.53049 −0.480231 −0.240116 − 0.970744i \(-0.577185\pi\)
−0.240116 + 0.970744i \(0.577185\pi\)
\(90\) 0 0
\(91\) −8.87270 −0.930112
\(92\) −5.73086 −0.597483
\(93\) 0 0
\(94\) −2.17202 −0.224027
\(95\) 0 0
\(96\) 0 0
\(97\) −10.7693 −1.09346 −0.546730 − 0.837309i \(-0.684128\pi\)
−0.546730 + 0.837309i \(0.684128\pi\)
\(98\) −11.5531 −1.16704
\(99\) 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.1.4 20
3.2 odd 2 inner 9675.2.a.da.1.17 20
5.2 odd 4 1935.2.b.g.1549.8 yes 40
5.3 odd 4 1935.2.b.g.1549.34 yes 40
5.4 even 2 9675.2.a.db.1.17 20
15.2 even 4 1935.2.b.g.1549.33 yes 40
15.8 even 4 1935.2.b.g.1549.7 ✓ 40
15.14 odd 2 9675.2.a.db.1.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1935.2.b.g.1549.7 ✓ 40 15.8 even 4
1935.2.b.g.1549.8 yes 40 5.2 odd 4
1935.2.b.g.1549.33 yes 40 15.2 even 4
1935.2.b.g.1549.34 yes 40 5.3 odd 4
9675.2.a.da.1.4 20 1.1 even 1 trivial
9675.2.a.da.1.17 20 3.2 odd 2 inner
9675.2.a.db.1.4 20 15.14 odd 2
9675.2.a.db.1.17 20 5.4 even 2