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Properties

Label 9675.2.a.cb.1.4
Level $9675$
Weight $2$
Character 9675.1
Self dual yes
Analytic conductor $77.255$
Analytic rank $0$
Dimension $5$
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 = [5,0,0,0,0,0,0,-3,0,0,9,0,1] 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: \(5\)
Coefficient field: 5.5.24217.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 5x^{3} - x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1075)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.722813\) 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)
 
\(f(q)\) \(=\) \(q+0.722813 q^{2} -1.47754 q^{4} -0.754729 q^{7} -2.51361 q^{8} +1.36316 q^{11} +7.11818 q^{13} -0.545528 q^{14} +1.13821 q^{16} +2.27409 q^{17} +3.30131 q^{19} +0.985308 q^{22} -3.01498 q^{23} +5.14511 q^{26} +1.11514 q^{28} +9.17534 q^{29} -7.20959 q^{31} +5.84994 q^{32} +1.64374 q^{34} -0.480643 q^{37} +2.38623 q^{38} -1.96029 q^{41} +1.00000 q^{43} -2.01412 q^{44} -2.17927 q^{46} -5.74014 q^{47} -6.43038 q^{49} -10.5174 q^{52} +12.3818 q^{53} +1.89710 q^{56} +6.63205 q^{58} -6.62955 q^{59} +9.53738 q^{61} -5.21118 q^{62} +1.95198 q^{64} -8.25480 q^{67} -3.36006 q^{68} +8.70502 q^{71} -6.45527 q^{73} -0.347415 q^{74} -4.87782 q^{76} -1.02882 q^{77} -16.0502 q^{79} -1.41692 q^{82} +3.58875 q^{83} +0.722813 q^{86} -3.42645 q^{88} +12.4291 q^{89} -5.37230 q^{91} +4.45476 q^{92} -4.14904 q^{94} -3.08745 q^{97} -4.64796 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{8} + 9 q^{11} + q^{13} + 7 q^{14} - 2 q^{16} + 3 q^{17} - 11 q^{19} - 13 q^{22} + 5 q^{26} + 15 q^{28} + 22 q^{29} - 5 q^{31} + 4 q^{32} - 7 q^{34} - 7 q^{37} - 3 q^{38} + 21 q^{41} + 5 q^{43}+ \cdots + 25 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.722813 0.511106 0.255553 − 0.966795i \(-0.417743\pi\)
0.255553 + 0.966795i \(0.417743\pi\)
\(3\) 0 0
\(4\) −1.47754 −0.738771
\(5\) 0 0
\(6\) 0 0
\(7\) −0.754729 −0.285261 −0.142630 − 0.989776i \(-0.545556\pi\)
−0.142630 + 0.989776i \(0.545556\pi\)
\(8\) −2.51361 −0.888696
\(9\) 0 0
\(10\) 0 0
\(11\) 1.36316 0.411008 0.205504 − 0.978656i \(-0.434117\pi\)
0.205504 + 0.978656i \(0.434117\pi\)
\(12\) 0 0
\(13\) 7.11818 1.97423 0.987114 − 0.160019i \(-0.0511557\pi\)
0.987114 + 0.160019i \(0.0511557\pi\)
\(14\) −0.545528 −0.145798
\(15\) 0 0
\(16\) 1.13821 0.284553
\(17\) 2.27409 0.551547 0.275773 − 0.961223i \(-0.411066\pi\)
0.275773 + 0.961223i \(0.411066\pi\)
\(18\) 0 0
\(19\) 3.30131 0.757372 0.378686 − 0.925525i \(-0.376376\pi\)
0.378686 + 0.925525i \(0.376376\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.985308 0.210068
\(23\) −3.01498 −0.628667 −0.314334 − 0.949313i \(-0.601781\pi\)
−0.314334 + 0.949313i \(0.601781\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.14511 1.00904
\(27\) 0 0
\(28\) 1.11514 0.210742
\(29\) 9.17534 1.70382 0.851908 − 0.523691i \(-0.175445\pi\)
0.851908 + 0.523691i \(0.175445\pi\)
\(30\) 0 0
\(31\) −7.20959 −1.29488 −0.647440 − 0.762116i \(-0.724161\pi\)
−0.647440 + 0.762116i \(0.724161\pi\)
\(32\) 5.84994 1.03413
\(33\) 0 0
\(34\) 1.64374 0.281899
\(35\) 0 0
\(36\) 0 0
\(37\) −0.480643 −0.0790172 −0.0395086 − 0.999219i \(-0.512579\pi\)
−0.0395086 + 0.999219i \(0.512579\pi\)
\(38\) 2.38623 0.387097
\(39\) 0 0
\(40\) 0 0
\(41\) −1.96029 −0.306146 −0.153073 − 0.988215i \(-0.548917\pi\)
−0.153073 + 0.988215i \(0.548917\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499
\(44\) −2.01412 −0.303641
\(45\) 0 0
\(46\) −2.17927 −0.321316
\(47\) −5.74014 −0.837285 −0.418642 − 0.908151i \(-0.637494\pi\)
−0.418642 + 0.908151i \(0.637494\pi\)
\(48\) 0 0
\(49\) −6.43038 −0.918626
\(50\) 0 0
\(51\) 0 0
\(52\) −10.5174 −1.45850
\(53\) 12.3818 1.70077 0.850387 − 0.526157i \(-0.176368\pi\)
0.850387 + 0.526157i \(0.176368\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.89710 0.253510
\(57\) 0 0
\(58\) 6.63205 0.870831
\(59\) −6.62955 −0.863094 −0.431547 − 0.902090i \(-0.642032\pi\)
−0.431547 + 0.902090i \(0.642032\pi\)
\(60\) 0 0
\(61\) 9.53738 1.22114 0.610568 − 0.791964i \(-0.290941\pi\)
0.610568 + 0.791964i \(0.290941\pi\)
\(62\) −5.21118 −0.661821
\(63\) 0 0
\(64\) 1.95198 0.243998
\(65\) 0 0
\(66\) 0 0
\(67\) −8.25480 −1.00848 −0.504242 − 0.863562i \(-0.668228\pi\)
−0.504242 + 0.863562i \(0.668228\pi\)
\(68\) −3.36006 −0.407467
\(69\) 0 0
\(70\) 0 0
\(71\) 8.70502 1.03310 0.516548 − 0.856258i \(-0.327217\pi\)
0.516548 + 0.856258i \(0.327217\pi\)
\(72\) 0 0
\(73\) −6.45527 −0.755532 −0.377766 − 0.925901i \(-0.623308\pi\)
−0.377766 + 0.925901i \(0.623308\pi\)
\(74\) −0.347415 −0.0403861
\(75\) 0 0
\(76\) −4.87782 −0.559525
\(77\) −1.02882 −0.117244
\(78\) 0 0
\(79\) −16.0502 −1.80579 −0.902896 − 0.429859i \(-0.858563\pi\)
−0.902896 + 0.429859i \(0.858563\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.41692 −0.156473
\(83\) 3.58875 0.393917 0.196958 − 0.980412i \(-0.436894\pi\)
0.196958 + 0.980412i \(0.436894\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.722813 0.0779429
\(87\) 0 0
\(88\) −3.42645 −0.365261
\(89\) 12.4291 1.31748 0.658741 − 0.752370i \(-0.271089\pi\)
0.658741 + 0.752370i \(0.271089\pi\)
\(90\) 0 0
\(91\) −5.37230 −0.563170
\(92\) 4.45476 0.464441
\(93\) 0 0
\(94\) −4.14904 −0.427941
\(95\) 0 0
\(96\) 0 0
\(97\) −3.08745 −0.313483 −0.156742 − 0.987640i \(-0.550099\pi\)
−0.156742 + 0.987640i \(0.550099\pi\)
\(98\) −4.64796 −0.469515
\(99\) 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.cb.1.4 5
3.2 odd 2 1075.2.a.o.1.2 yes 5
5.4 even 2 9675.2.a.cc.1.2 5
15.2 even 4 1075.2.b.i.474.4 10
15.8 even 4 1075.2.b.i.474.7 10
15.14 odd 2 1075.2.a.n.1.4 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1075.2.a.n.1.4 ✓ 5 15.14 odd 2
1075.2.a.o.1.2 yes 5 3.2 odd 2
1075.2.b.i.474.4 10 15.2 even 4
1075.2.b.i.474.7 10 15.8 even 4
9675.2.a.cb.1.4 5 1.1 even 1 trivial
9675.2.a.cc.1.2 5 5.4 even 2