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Properties

Label 9300.2.a.t.1.3
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 1.3
Root \(4.47467\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.47467 q^{7} +1.00000 q^{9} +2.00000 q^{11} -4.47467 q^{13} -1.07331 q^{17} +4.00000 q^{19} -4.47467 q^{21} +1.07331 q^{23} -1.00000 q^{27} -3.54798 q^{29} -1.00000 q^{31} -2.00000 q^{33} -2.47467 q^{37} +4.47467 q^{39} +4.00000 q^{41} -2.00000 q^{43} +8.02265 q^{47} +13.0226 q^{49} +1.07331 q^{51} -10.0226 q^{53} -4.00000 q^{57} +9.54798 q^{59} -0.146623 q^{61} +4.47467 q^{63} +4.32804 q^{67} -1.07331 q^{69} +12.4973 q^{71} -15.4240 q^{73} +8.94933 q^{77} +9.87602 q^{79} +1.00000 q^{81} +2.92669 q^{83} +3.54798 q^{87} +15.4014 q^{89} -20.0226 q^{91} +1.00000 q^{93} -1.85338 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{9} + 6 q^{11} - 2 q^{17} + 12 q^{19} + 2 q^{23} - 3 q^{27} + 4 q^{29} - 3 q^{31} - 6 q^{33} + 6 q^{37} + 12 q^{41} - 6 q^{43} - 4 q^{47} + 11 q^{49} + 2 q^{51} - 2 q^{53} - 12 q^{57}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(7\) 4.47467 1.69127 0.845633 − 0.533765i \(-0.179223\pi\)
0.845633 + 0.533765i \(0.179223\pi\)
\(8\) 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
\(13\) −4.47467 −1.24105 −0.620525 − 0.784187i \(-0.713080\pi\)
−0.620525 + 0.784187i \(0.713080\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.07331 −0.260316 −0.130158 − 0.991493i \(-0.541549\pi\)
−0.130158 + 0.991493i \(0.541549\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −4.47467 −0.976452
\(22\) 0 0
\(23\) 1.07331 0.223801 0.111900 − 0.993719i \(-0.464306\pi\)
0.111900 + 0.993719i \(0.464306\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.54798 −0.658843 −0.329422 − 0.944183i \(-0.606854\pi\)
−0.329422 + 0.944183i \(0.606854\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.47467 −0.406833 −0.203416 − 0.979092i \(-0.565205\pi\)
−0.203416 + 0.979092i \(0.565205\pi\)
\(38\) 0 0
\(39\) 4.47467 0.716520
\(40\) 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
\(47\) 8.02265 1.17022 0.585112 − 0.810953i \(-0.301051\pi\)
0.585112 + 0.810953i \(0.301051\pi\)
\(48\) 0 0
\(49\) 13.0226 1.86038
\(50\) 0 0
\(51\) 1.07331 0.150294
\(52\) 0 0
\(53\) −10.0226 −1.37672 −0.688358 − 0.725371i \(-0.741668\pi\)
−0.688358 + 0.725371i \(0.741668\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 9.54798 1.24304 0.621520 − 0.783398i \(-0.286515\pi\)
0.621520 + 0.783398i \(0.286515\pi\)
\(60\) 0 0
\(61\) −0.146623 −0.0187731 −0.00938657 − 0.999956i \(-0.502988\pi\)
−0.00938657 + 0.999956i \(0.502988\pi\)
\(62\) 0 0
\(63\) 4.47467 0.563755
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.32804 0.528755 0.264377 − 0.964419i \(-0.414834\pi\)
0.264377 + 0.964419i \(0.414834\pi\)
\(68\) 0 0
\(69\) −1.07331 −0.129212
\(70\) 0 0
\(71\) 12.4973 1.48316 0.741579 − 0.670865i \(-0.234077\pi\)
0.741579 + 0.670865i \(0.234077\pi\)
\(72\) 0 0
\(73\) −15.4240 −1.80524 −0.902621 − 0.430435i \(-0.858360\pi\)
−0.902621 + 0.430435i \(0.858360\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.94933 1.01987
\(78\) 0 0
\(79\) 9.87602 1.11114 0.555570 − 0.831470i \(-0.312500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.92669 0.321246 0.160623 − 0.987016i \(-0.448650\pi\)
0.160623 + 0.987016i \(0.448650\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.54798 0.380383
\(88\) 0 0
\(89\) 15.4014 1.63254 0.816270 − 0.577670i \(-0.196038\pi\)
0.816270 + 0.577670i \(0.196038\pi\)
\(90\) 0 0
\(91\) −20.0226 −2.09894
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.85338 −0.188182 −0.0940910 − 0.995564i \(-0.529994\pi\)
−0.0940910 + 0.995564i \(0.529994\pi\)
\(98\) 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.1.3 3
5.2 odd 4 9300.2.g.r.3349.6 6
5.3 odd 4 9300.2.g.r.3349.1 6
5.4 even 2 1860.2.a.h.1.1 ✓ 3
15.14 odd 2 5580.2.a.i.1.1 3
20.19 odd 2 7440.2.a.bq.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.a.h.1.1 ✓ 3 5.4 even 2
5580.2.a.i.1.1 3 15.14 odd 2
7440.2.a.bq.1.3 3 20.19 odd 2
9300.2.a.t.1.3 3 1.1 even 1 trivial
9300.2.g.r.3349.1 6 5.3 odd 4
9300.2.g.r.3349.6 6 5.2 odd 4