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Properties

Label 7440.2.a.bw.1.1
Level $7440$
Weight $2$
Character 7440.1
Self dual yes
Analytic conductor $59.409$
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] = [7440,2,Mod(1,7440)] 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("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7440.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,0,3,0,-1,0,3,0,-5,0,-2,0,3,0,0] 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: \(59.4086991038\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3720)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 7440.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -3.12489 q^{7} +1.00000 q^{9} -1.51514 q^{11} -3.60975 q^{13} +1.00000 q^{15} +5.28005 q^{17} -1.51514 q^{19} -3.12489 q^{21} +1.76491 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.60975 q^{29} -1.00000 q^{31} -1.51514 q^{33} -3.12489 q^{35} +4.57947 q^{37} -3.60975 q^{39} +0.249771 q^{41} -10.7952 q^{43} +1.00000 q^{45} +6.49954 q^{47} +2.76491 q^{49} +5.28005 q^{51} +3.45459 q^{53} -1.51514 q^{55} -1.51514 q^{57} -8.88979 q^{59} -7.28005 q^{61} -3.12489 q^{63} -3.60975 q^{65} -6.57947 q^{67} +1.76491 q^{69} +15.6850 q^{71} -8.34438 q^{73} +1.00000 q^{75} +4.73463 q^{77} -5.45459 q^{79} +1.00000 q^{81} -6.96972 q^{83} +5.28005 q^{85} +1.60975 q^{87} +2.15516 q^{89} +11.2800 q^{91} -1.00000 q^{93} -1.51514 q^{95} -6.49954 q^{97} -1.51514 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} - 5 q^{11} - 2 q^{13} + 3 q^{15} - 5 q^{19} - q^{21} - 11 q^{23} + 3 q^{25} + 3 q^{27} - 4 q^{29} - 3 q^{31} - 5 q^{33} - q^{35} + 4 q^{37} - 2 q^{39} - 16 q^{41}+ \cdots - 5 q^{99}+O(q^{100}) \) 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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.12489 −1.18110 −0.590548 − 0.807003i \(-0.701088\pi\)
−0.590548 + 0.807003i \(0.701088\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.51514 −0.456831 −0.228416 − 0.973564i \(-0.573355\pi\)
−0.228416 + 0.973564i \(0.573355\pi\)
\(12\) 0 0
\(13\) −3.60975 −1.00116 −0.500582 − 0.865689i \(-0.666881\pi\)
−0.500582 + 0.865689i \(0.666881\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 5.28005 1.28060 0.640300 − 0.768125i \(-0.278810\pi\)
0.640300 + 0.768125i \(0.278810\pi\)
\(18\) 0 0
\(19\) −1.51514 −0.347597 −0.173798 − 0.984781i \(-0.555604\pi\)
−0.173798 + 0.984781i \(0.555604\pi\)
\(20\) 0 0
\(21\) −3.12489 −0.681906
\(22\) 0 0
\(23\) 1.76491 0.368009 0.184004 − 0.982925i \(-0.441094\pi\)
0.184004 + 0.982925i \(0.441094\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.60975 0.298923 0.149461 − 0.988768i \(-0.452246\pi\)
0.149461 + 0.988768i \(0.452246\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −1.51514 −0.263752
\(34\) 0 0
\(35\) −3.12489 −0.528202
\(36\) 0 0
\(37\) 4.57947 0.752860 0.376430 − 0.926445i \(-0.377151\pi\)
0.376430 + 0.926445i \(0.377151\pi\)
\(38\) 0 0
\(39\) −3.60975 −0.578022
\(40\) 0 0
\(41\) 0.249771 0.0390077 0.0195038 − 0.999810i \(-0.493791\pi\)
0.0195038 + 0.999810i \(0.493791\pi\)
\(42\) 0 0
\(43\) −10.7952 −1.64625 −0.823125 − 0.567860i \(-0.807771\pi\)
−0.823125 + 0.567860i \(0.807771\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.49954 0.948056 0.474028 − 0.880510i \(-0.342800\pi\)
0.474028 + 0.880510i \(0.342800\pi\)
\(48\) 0 0
\(49\) 2.76491 0.394987
\(50\) 0 0
\(51\) 5.28005 0.739354
\(52\) 0 0
\(53\) 3.45459 0.474524 0.237262 − 0.971446i \(-0.423750\pi\)
0.237262 + 0.971446i \(0.423750\pi\)
\(54\) 0 0
\(55\) −1.51514 −0.204301
\(56\) 0 0
\(57\) −1.51514 −0.200685
\(58\) 0 0
\(59\) −8.88979 −1.15735 −0.578676 − 0.815557i \(-0.696431\pi\)
−0.578676 + 0.815557i \(0.696431\pi\)
\(60\) 0 0
\(61\) −7.28005 −0.932114 −0.466057 − 0.884755i \(-0.654326\pi\)
−0.466057 + 0.884755i \(0.654326\pi\)
\(62\) 0 0
\(63\) −3.12489 −0.393699
\(64\) 0 0
\(65\) −3.60975 −0.447734
\(66\) 0 0
\(67\) −6.57947 −0.803810 −0.401905 − 0.915681i \(-0.631652\pi\)
−0.401905 + 0.915681i \(0.631652\pi\)
\(68\) 0 0
\(69\) 1.76491 0.212470
\(70\) 0 0
\(71\) 15.6850 1.86146 0.930732 − 0.365701i \(-0.119171\pi\)
0.930732 + 0.365701i \(0.119171\pi\)
\(72\) 0 0
\(73\) −8.34438 −0.976636 −0.488318 − 0.872666i \(-0.662389\pi\)
−0.488318 + 0.872666i \(0.662389\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 4.73463 0.539561
\(78\) 0 0
\(79\) −5.45459 −0.613689 −0.306844 − 0.951760i \(-0.599273\pi\)
−0.306844 + 0.951760i \(0.599273\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.96972 −0.765027 −0.382513 − 0.923950i \(-0.624941\pi\)
−0.382513 + 0.923950i \(0.624941\pi\)
\(84\) 0 0
\(85\) 5.28005 0.572701
\(86\) 0 0
\(87\) 1.60975 0.172583
\(88\) 0 0
\(89\) 2.15516 0.228447 0.114223 − 0.993455i \(-0.463562\pi\)
0.114223 + 0.993455i \(0.463562\pi\)
\(90\) 0 0
\(91\) 11.2800 1.18247
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −1.51514 −0.155450
\(96\) 0 0
\(97\) −6.49954 −0.659928 −0.329964 − 0.943993i \(-0.607037\pi\)
−0.329964 + 0.943993i \(0.607037\pi\)
\(98\) 0 0
\(99\) −1.51514 −0.152277
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 7440.2.a.bw.1.1 3
4.3 odd 2 3720.2.a.m.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.m.1.3 ✓ 3 4.3 odd 2
7440.2.a.bw.1.1 3 1.1 even 1 trivial