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Properties

Label 9300.2.a.t.1.2
Level $9300$
Weight $2$
Character 9300.1
Self dual yes
Analytic conductor $74.261$
Analytic rank $0$
Dimension $3$
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] = [9300,2,Mod(1,9300)] 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("9300.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,0,0,0,0,3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(74.2608738798\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.7636.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 16x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1860)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.24586\) of defining polynomial
Character \(\chi\) \(=\) 9300.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.24586 q^{7} +1.00000 q^{9} +2.00000 q^{11} +1.24586 q^{13} +5.95610 q^{17} +4.00000 q^{19} +1.24586 q^{21} -5.95610 q^{23} -1.00000 q^{27} +9.20197 q^{29} -1.00000 q^{31} -2.00000 q^{33} +3.24586 q^{37} -1.24586 q^{39} +4.00000 q^{41} -2.00000 q^{43} -10.4478 q^{47} -5.44783 q^{49} -5.95610 q^{51} +8.44783 q^{53} -4.00000 q^{57} -3.20197 q^{59} +13.9122 q^{61} -1.24586 q^{63} +12.6663 q^{67} +5.95610 q^{69} -11.6937 q^{71} +1.73759 q^{73} -2.49172 q^{77} +5.46438 q^{79} +1.00000 q^{81} +9.95610 q^{83} -9.20197 q^{87} +16.7102 q^{89} -1.55217 q^{91} +1.00000 q^{93} -15.9122 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{9} + 6 q^{11} - 2 q^{17} + 12 q^{19} + 2 q^{23} - 3 q^{27} + 4 q^{29} - 3 q^{31} - 6 q^{33} + 6 q^{37} + 12 q^{41} - 6 q^{43} - 4 q^{47} + 11 q^{49} + 2 q^{51} - 2 q^{53} - 12 q^{57}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.24586 −0.470892 −0.235446 − 0.971887i \(-0.575655\pi\)
−0.235446 + 0.971887i \(0.575655\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 1.24586 0.345540 0.172770 − 0.984962i \(-0.444728\pi\)
0.172770 + 0.984962i \(0.444728\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.95610 1.44457 0.722284 − 0.691597i \(-0.243093\pi\)
0.722284 + 0.691597i \(0.243093\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 1.24586 0.271869
\(22\) 0 0
\(23\) −5.95610 −1.24193 −0.620967 − 0.783837i \(-0.713260\pi\)
−0.620967 + 0.783837i \(0.713260\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 9.20197 1.70876 0.854381 − 0.519647i \(-0.173937\pi\)
0.854381 + 0.519647i \(0.173937\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.24586 0.533616 0.266808 − 0.963750i \(-0.414031\pi\)
0.266808 + 0.963750i \(0.414031\pi\)
\(38\) 0 0
\(39\) −1.24586 −0.199498
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.4478 −1.52397 −0.761986 − 0.647593i \(-0.775776\pi\)
−0.761986 + 0.647593i \(0.775776\pi\)
\(48\) 0 0
\(49\) −5.44783 −0.778261
\(50\) 0 0
\(51\) −5.95610 −0.834021
\(52\) 0 0
\(53\) 8.44783 1.16040 0.580199 − 0.814475i \(-0.302975\pi\)
0.580199 + 0.814475i \(0.302975\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −3.20197 −0.416860 −0.208430 − 0.978037i \(-0.566835\pi\)
−0.208430 + 0.978037i \(0.566835\pi\)
\(60\) 0 0
\(61\) 13.9122 1.78128 0.890638 − 0.454713i \(-0.150258\pi\)
0.890638 + 0.454713i \(0.150258\pi\)
\(62\) 0 0
\(63\) −1.24586 −0.156964
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.6663 1.54744 0.773720 − 0.633527i \(-0.218394\pi\)
0.773720 + 0.633527i \(0.218394\pi\)
\(68\) 0 0
\(69\) 5.95610 0.717031
\(70\) 0 0
\(71\) −11.6937 −1.38779 −0.693893 − 0.720078i \(-0.744106\pi\)
−0.693893 + 0.720078i \(0.744106\pi\)
\(72\) 0 0
\(73\) 1.73759 0.203369 0.101685 − 0.994817i \(-0.467577\pi\)
0.101685 + 0.994817i \(0.467577\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.49172 −0.283958
\(78\) 0 0
\(79\) 5.46438 0.614791 0.307395 − 0.951582i \(-0.400543\pi\)
0.307395 + 0.951582i \(0.400543\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.95610 1.09282 0.546412 − 0.837516i \(-0.315993\pi\)
0.546412 + 0.837516i \(0.315993\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.20197 −0.986554
\(88\) 0 0
\(89\) 16.7102 1.77128 0.885641 − 0.464370i \(-0.153719\pi\)
0.885641 + 0.464370i \(0.153719\pi\)
\(90\) 0 0
\(91\) −1.55217 −0.162712
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −15.9122 −1.61564 −0.807820 − 0.589429i \(-0.799353\pi\)
−0.807820 + 0.589429i \(0.799353\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
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 9300.2.a.t.1.2 3
5.2 odd 4 9300.2.g.r.3349.5 6
5.3 odd 4 9300.2.g.r.3349.2 6
5.4 even 2 1860.2.a.h.1.2 ✓ 3
15.14 odd 2 5580.2.a.i.1.2 3
20.19 odd 2 7440.2.a.bq.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.a.h.1.2 ✓ 3 5.4 even 2
5580.2.a.i.1.2 3 15.14 odd 2
7440.2.a.bq.1.2 3 20.19 odd 2
9300.2.a.t.1.2 3 1.1 even 1 trivial
9300.2.g.r.3349.2 6 5.3 odd 4
9300.2.g.r.3349.5 6 5.2 odd 4