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Properties

Label 9072.2.a.bq.1.1
Level $9072$
Weight $2$
Character 9072.1
Self dual yes
Analytic conductor $72.440$
Analytic rank $1$
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] = [9072,2,Mod(1,9072)] 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("9072.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9072, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-5,0,3,0,0,0,2,0,3,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.4402847137\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.69963\) of defining polynomial
Character \(\chi\) \(=\) 9072.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.58836 q^{5} +1.00000 q^{7} +2.81089 q^{11} +1.00000 q^{13} -4.11126 q^{17} +0.888736 q^{19} -5.87636 q^{23} +7.87636 q^{25} -1.69963 q^{29} +6.98762 q^{31} -3.58836 q^{35} +4.76509 q^{37} -5.41164 q^{41} -5.21015 q^{43} +2.66621 q^{47} +1.00000 q^{49} -0.123644 q^{53} -10.0865 q^{55} +8.87636 q^{59} +3.87636 q^{61} -3.58836 q^{65} -12.3090 q^{67} +2.87636 q^{71} -10.6414 q^{73} +2.81089 q^{77} +7.08650 q^{79} +4.11126 q^{83} +14.7527 q^{85} +9.60940 q^{89} +1.00000 q^{91} -3.18911 q^{95} +7.32141 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 5 q^{5} + 3 q^{7} + 2 q^{11} + 3 q^{13} - 12 q^{17} + 3 q^{19} + 6 q^{25} + q^{29} + 3 q^{31} - 5 q^{35} - 3 q^{37} - 22 q^{41} + 3 q^{43} + 9 q^{47} + 3 q^{49} - 18 q^{53} + 6 q^{55} + 9 q^{59}+ \cdots + 3 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −3.58836 −1.60477 −0.802383 − 0.596810i \(-0.796435\pi\)
−0.802383 + 0.596810i \(0.796435\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.81089 0.847516 0.423758 − 0.905775i \(-0.360711\pi\)
0.423758 + 0.905775i \(0.360711\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.11126 −0.997128 −0.498564 − 0.866853i \(-0.666139\pi\)
−0.498564 + 0.866853i \(0.666139\pi\)
\(18\) 0 0
\(19\) 0.888736 0.203890 0.101945 − 0.994790i \(-0.467493\pi\)
0.101945 + 0.994790i \(0.467493\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.87636 −1.22530 −0.612652 − 0.790352i \(-0.709897\pi\)
−0.612652 + 0.790352i \(0.709897\pi\)
\(24\) 0 0
\(25\) 7.87636 1.57527
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.69963 −0.315613 −0.157807 − 0.987470i \(-0.550442\pi\)
−0.157807 + 0.987470i \(0.550442\pi\)
\(30\) 0 0
\(31\) 6.98762 1.25501 0.627507 − 0.778611i \(-0.284075\pi\)
0.627507 + 0.778611i \(0.284075\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.58836 −0.606544
\(36\) 0 0
\(37\) 4.76509 0.783376 0.391688 − 0.920098i \(-0.371891\pi\)
0.391688 + 0.920098i \(0.371891\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.41164 −0.845156 −0.422578 − 0.906327i \(-0.638875\pi\)
−0.422578 + 0.906327i \(0.638875\pi\)
\(42\) 0 0
\(43\) −5.21015 −0.794540 −0.397270 − 0.917702i \(-0.630042\pi\)
−0.397270 + 0.917702i \(0.630042\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.66621 0.388906 0.194453 − 0.980912i \(-0.437707\pi\)
0.194453 + 0.980912i \(0.437707\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.123644 −0.0169838 −0.00849190 − 0.999964i \(-0.502703\pi\)
−0.00849190 + 0.999964i \(0.502703\pi\)
\(54\) 0 0
\(55\) −10.0865 −1.36006
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.87636 1.15560 0.577802 − 0.816177i \(-0.303911\pi\)
0.577802 + 0.816177i \(0.303911\pi\)
\(60\) 0 0
\(61\) 3.87636 0.496317 0.248158 − 0.968720i \(-0.420175\pi\)
0.248158 + 0.968720i \(0.420175\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.58836 −0.445082
\(66\) 0 0
\(67\) −12.3090 −1.50379 −0.751894 − 0.659284i \(-0.770859\pi\)
−0.751894 + 0.659284i \(0.770859\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.87636 0.341361 0.170680 − 0.985326i \(-0.445403\pi\)
0.170680 + 0.985326i \(0.445403\pi\)
\(72\) 0 0
\(73\) −10.6414 −1.24549 −0.622744 − 0.782426i \(-0.713982\pi\)
−0.622744 + 0.782426i \(0.713982\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.81089 0.320331
\(78\) 0 0
\(79\) 7.08650 0.797294 0.398647 − 0.917104i \(-0.369480\pi\)
0.398647 + 0.917104i \(0.369480\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.11126 0.451270 0.225635 − 0.974212i \(-0.427554\pi\)
0.225635 + 0.974212i \(0.427554\pi\)
\(84\) 0 0
\(85\) 14.7527 1.60016
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.60940 1.01859 0.509297 − 0.860591i \(-0.329905\pi\)
0.509297 + 0.860591i \(0.329905\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.18911 −0.327196
\(96\) 0 0
\(97\) 7.32141 0.743377 0.371688 − 0.928358i \(-0.378779\pi\)
0.371688 + 0.928358i \(0.378779\pi\)
\(98\) 0 0
\(99\) 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 9072.2.a.bq.1.1 3
3.2 odd 2 9072.2.a.cd.1.3 3
4.3 odd 2 567.2.a.d.1.3 3
9.2 odd 6 3024.2.r.g.1009.1 6
9.4 even 3 1008.2.r.k.673.1 6
9.5 odd 6 3024.2.r.g.2017.1 6
9.7 even 3 1008.2.r.k.337.1 6
12.11 even 2 567.2.a.g.1.1 3
28.27 even 2 3969.2.a.m.1.3 3
36.7 odd 6 63.2.f.b.22.1 ✓ 6
36.11 even 6 189.2.f.a.64.3 6
36.23 even 6 189.2.f.a.127.3 6
36.31 odd 6 63.2.f.b.43.1 yes 6
84.83 odd 2 3969.2.a.p.1.1 3
252.11 even 6 1323.2.h.d.226.1 6
252.23 even 6 1323.2.h.d.802.1 6
252.31 even 6 441.2.g.d.79.1 6
252.47 odd 6 1323.2.g.b.361.3 6
252.59 odd 6 1323.2.g.b.667.3 6
252.67 odd 6 441.2.g.e.79.1 6
252.79 odd 6 441.2.g.e.67.1 6
252.83 odd 6 1323.2.f.c.442.3 6
252.95 even 6 1323.2.g.c.667.3 6
252.103 even 6 441.2.h.b.214.3 6
252.115 even 6 441.2.h.b.373.3 6
252.131 odd 6 1323.2.h.e.802.1 6
252.139 even 6 441.2.f.d.295.1 6
252.151 odd 6 441.2.h.c.373.3 6
252.167 odd 6 1323.2.f.c.883.3 6
252.187 even 6 441.2.g.d.67.1 6
252.191 even 6 1323.2.g.c.361.3 6
252.223 even 6 441.2.f.d.148.1 6
252.227 odd 6 1323.2.h.e.226.1 6
252.247 odd 6 441.2.h.c.214.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.1 ✓ 6 36.7 odd 6
63.2.f.b.43.1 yes 6 36.31 odd 6
189.2.f.a.64.3 6 36.11 even 6
189.2.f.a.127.3 6 36.23 even 6
441.2.f.d.148.1 6 252.223 even 6
441.2.f.d.295.1 6 252.139 even 6
441.2.g.d.67.1 6 252.187 even 6
441.2.g.d.79.1 6 252.31 even 6
441.2.g.e.67.1 6 252.79 odd 6
441.2.g.e.79.1 6 252.67 odd 6
441.2.h.b.214.3 6 252.103 even 6
441.2.h.b.373.3 6 252.115 even 6
441.2.h.c.214.3 6 252.247 odd 6
441.2.h.c.373.3 6 252.151 odd 6
567.2.a.d.1.3 3 4.3 odd 2
567.2.a.g.1.1 3 12.11 even 2
1008.2.r.k.337.1 6 9.7 even 3
1008.2.r.k.673.1 6 9.4 even 3
1323.2.f.c.442.3 6 252.83 odd 6
1323.2.f.c.883.3 6 252.167 odd 6
1323.2.g.b.361.3 6 252.47 odd 6
1323.2.g.b.667.3 6 252.59 odd 6
1323.2.g.c.361.3 6 252.191 even 6
1323.2.g.c.667.3 6 252.95 even 6
1323.2.h.d.226.1 6 252.11 even 6
1323.2.h.d.802.1 6 252.23 even 6
1323.2.h.e.226.1 6 252.227 odd 6
1323.2.h.e.802.1 6 252.131 odd 6
3024.2.r.g.1009.1 6 9.2 odd 6
3024.2.r.g.2017.1 6 9.5 odd 6
3969.2.a.m.1.3 3 28.27 even 2
3969.2.a.p.1.1 3 84.83 odd 2
9072.2.a.bq.1.1 3 1.1 even 1 trivial
9072.2.a.cd.1.3 3 3.2 odd 2