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Properties

Label 7440.2.a.bw.1.2
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 1.2
Root \(-0.363328\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +0.363328 q^{7} +1.00000 q^{9} +1.14134 q^{11} -2.77801 q^{13} +1.00000 q^{15} -7.00933 q^{17} +1.14134 q^{19} +0.363328 q^{21} -7.86799 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.778008 q^{29} -1.00000 q^{31} +1.14134 q^{33} +0.363328 q^{35} +9.06068 q^{37} -2.77801 q^{39} -6.72666 q^{41} +4.15066 q^{43} +1.00000 q^{45} -7.45331 q^{47} -6.86799 q^{49} -7.00933 q^{51} +11.4240 q^{53} +1.14134 q^{55} +1.14134 q^{57} +4.23132 q^{59} +5.00933 q^{61} +0.363328 q^{63} -2.77801 q^{65} -11.0607 q^{67} -7.86799 q^{69} -12.3820 q^{71} -3.19269 q^{73} +1.00000 q^{75} +0.414680 q^{77} -13.4240 q^{79} +1.00000 q^{81} -12.2827 q^{83} -7.00933 q^{85} +0.778008 q^{87} -6.64600 q^{89} -1.00933 q^{91} -1.00000 q^{93} +1.14134 q^{95} +7.45331 q^{97} +1.14134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} - 5 q^{11} - 2 q^{13} + 3 q^{15} - 5 q^{19} - q^{21} - 11 q^{23} + 3 q^{25} + 3 q^{27} - 4 q^{29} - 3 q^{31} - 5 q^{33} - q^{35} + 4 q^{37} - 2 q^{39} - 16 q^{41}+ \cdots - 5 q^{99}+O(q^{100}) \) 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
\(7\) 0.363328 0.137325 0.0686626 − 0.997640i \(-0.478127\pi\)
0.0686626 + 0.997640i \(0.478127\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.14134 0.344126 0.172063 − 0.985086i \(-0.444957\pi\)
0.172063 + 0.985086i \(0.444957\pi\)
\(12\) 0 0
\(13\) −2.77801 −0.770481 −0.385240 − 0.922816i \(-0.625881\pi\)
−0.385240 + 0.922816i \(0.625881\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −7.00933 −1.70001 −0.850006 − 0.526773i \(-0.823402\pi\)
−0.850006 + 0.526773i \(0.823402\pi\)
\(18\) 0 0
\(19\) 1.14134 0.261840 0.130920 − 0.991393i \(-0.458207\pi\)
0.130920 + 0.991393i \(0.458207\pi\)
\(20\) 0 0
\(21\) 0.363328 0.0792847
\(22\) 0 0
\(23\) −7.86799 −1.64059 −0.820295 − 0.571941i \(-0.806191\pi\)
−0.820295 + 0.571941i \(0.806191\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.778008 0.144472 0.0722362 − 0.997388i \(-0.476986\pi\)
0.0722362 + 0.997388i \(0.476986\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) 1.14134 0.198681
\(34\) 0 0
\(35\) 0.363328 0.0614137
\(36\) 0 0
\(37\) 9.06068 1.48957 0.744783 − 0.667306i \(-0.232553\pi\)
0.744783 + 0.667306i \(0.232553\pi\)
\(38\) 0 0
\(39\) −2.77801 −0.444837
\(40\) 0 0
\(41\) −6.72666 −1.05053 −0.525264 − 0.850940i \(-0.676033\pi\)
−0.525264 + 0.850940i \(0.676033\pi\)
\(42\) 0 0
\(43\) 4.15066 0.632970 0.316485 − 0.948597i \(-0.397497\pi\)
0.316485 + 0.948597i \(0.397497\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −7.45331 −1.08718 −0.543589 − 0.839352i \(-0.682935\pi\)
−0.543589 + 0.839352i \(0.682935\pi\)
\(48\) 0 0
\(49\) −6.86799 −0.981142
\(50\) 0 0
\(51\) −7.00933 −0.981502
\(52\) 0 0
\(53\) 11.4240 1.56921 0.784604 − 0.619997i \(-0.212866\pi\)
0.784604 + 0.619997i \(0.212866\pi\)
\(54\) 0 0
\(55\) 1.14134 0.153898
\(56\) 0 0
\(57\) 1.14134 0.151174
\(58\) 0 0
\(59\) 4.23132 0.550871 0.275436 − 0.961320i \(-0.411178\pi\)
0.275436 + 0.961320i \(0.411178\pi\)
\(60\) 0 0
\(61\) 5.00933 0.641379 0.320689 − 0.947184i \(-0.396085\pi\)
0.320689 + 0.947184i \(0.396085\pi\)
\(62\) 0 0
\(63\) 0.363328 0.0457751
\(64\) 0 0
\(65\) −2.77801 −0.344569
\(66\) 0 0
\(67\) −11.0607 −1.35128 −0.675639 − 0.737233i \(-0.736132\pi\)
−0.675639 + 0.737233i \(0.736132\pi\)
\(68\) 0 0
\(69\) −7.86799 −0.947195
\(70\) 0 0
\(71\) −12.3820 −1.46947 −0.734736 − 0.678354i \(-0.762694\pi\)
−0.734736 + 0.678354i \(0.762694\pi\)
\(72\) 0 0
\(73\) −3.19269 −0.373676 −0.186838 − 0.982391i \(-0.559824\pi\)
−0.186838 + 0.982391i \(0.559824\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0.414680 0.0472571
\(78\) 0 0
\(79\) −13.4240 −1.51032 −0.755159 − 0.655541i \(-0.772441\pi\)
−0.755159 + 0.655541i \(0.772441\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.2827 −1.34820 −0.674099 − 0.738641i \(-0.735468\pi\)
−0.674099 + 0.738641i \(0.735468\pi\)
\(84\) 0 0
\(85\) −7.00933 −0.760268
\(86\) 0 0
\(87\) 0.778008 0.0834112
\(88\) 0 0
\(89\) −6.64600 −0.704475 −0.352237 − 0.935911i \(-0.614579\pi\)
−0.352237 + 0.935911i \(0.614579\pi\)
\(90\) 0 0
\(91\) −1.00933 −0.105806
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 1.14134 0.117099
\(96\) 0 0
\(97\) 7.45331 0.756769 0.378385 − 0.925648i \(-0.376480\pi\)
0.378385 + 0.925648i \(0.376480\pi\)
\(98\) 0 0
\(99\) 1.14134 0.114709
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.1.2 3
4.3 odd 2 3720.2.a.m.1.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.m.1.2 ✓ 3 4.3 odd 2
7440.2.a.bw.1.2 3 1.1 even 1 trivial