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Properties

Label 7440.2.a.bw.1.3
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 1.3
Root \(-1.76156\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.76156 q^{7} +1.00000 q^{9} -4.62620 q^{11} +4.38776 q^{13} +1.00000 q^{15} +1.72928 q^{17} -4.62620 q^{19} +1.76156 q^{21} -4.89692 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.38776 q^{29} -1.00000 q^{31} -4.62620 q^{33} +1.76156 q^{35} -9.64015 q^{37} +4.38776 q^{39} -9.52311 q^{41} -10.3555 q^{43} +1.00000 q^{45} -13.0462 q^{47} -3.89692 q^{49} +1.72928 q^{51} -5.87859 q^{53} -4.62620 q^{55} -4.62620 q^{57} +2.65847 q^{59} -3.72928 q^{61} +1.76156 q^{63} +4.38776 q^{65} +7.64015 q^{67} -4.89692 q^{69} +3.69701 q^{71} +12.5371 q^{73} +1.00000 q^{75} -8.14931 q^{77} +3.87859 q^{79} +1.00000 q^{81} -0.747604 q^{83} +1.72928 q^{85} -6.38776 q^{87} +3.49084 q^{89} +7.72928 q^{91} -1.00000 q^{93} -4.62620 q^{95} +13.0462 q^{97} -4.62620 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} - 5 q^{11} - 2 q^{13} + 3 q^{15} - 5 q^{19} - q^{21} - 11 q^{23} + 3 q^{25} + 3 q^{27} - 4 q^{29} - 3 q^{31} - 5 q^{33} - q^{35} + 4 q^{37} - 2 q^{39} - 16 q^{41}+ \cdots - 5 q^{99}+O(q^{100}) \) 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
\(7\) 1.76156 0.665806 0.332903 − 0.942961i \(-0.391972\pi\)
0.332903 + 0.942961i \(0.391972\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.62620 −1.39485 −0.697426 − 0.716657i \(-0.745671\pi\)
−0.697426 + 0.716657i \(0.745671\pi\)
\(12\) 0 0
\(13\) 4.38776 1.21694 0.608472 − 0.793575i \(-0.291783\pi\)
0.608472 + 0.793575i \(0.291783\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 1.72928 0.419412 0.209706 − 0.977764i \(-0.432749\pi\)
0.209706 + 0.977764i \(0.432749\pi\)
\(18\) 0 0
\(19\) −4.62620 −1.06132 −0.530661 − 0.847584i \(-0.678056\pi\)
−0.530661 + 0.847584i \(0.678056\pi\)
\(20\) 0 0
\(21\) 1.76156 0.384403
\(22\) 0 0
\(23\) −4.89692 −1.02108 −0.510539 − 0.859855i \(-0.670554\pi\)
−0.510539 + 0.859855i \(0.670554\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.38776 −1.18618 −0.593088 − 0.805138i \(-0.702091\pi\)
−0.593088 + 0.805138i \(0.702091\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −4.62620 −0.805318
\(34\) 0 0
\(35\) 1.76156 0.297758
\(36\) 0 0
\(37\) −9.64015 −1.58483 −0.792416 − 0.609982i \(-0.791177\pi\)
−0.792416 + 0.609982i \(0.791177\pi\)
\(38\) 0 0
\(39\) 4.38776 0.702603
\(40\) 0 0
\(41\) −9.52311 −1.48726 −0.743630 − 0.668591i \(-0.766898\pi\)
−0.743630 + 0.668591i \(0.766898\pi\)
\(42\) 0 0
\(43\) −10.3555 −1.57920 −0.789598 − 0.613625i \(-0.789711\pi\)
−0.789598 + 0.613625i \(0.789711\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −13.0462 −1.90299 −0.951494 − 0.307667i \(-0.900452\pi\)
−0.951494 + 0.307667i \(0.900452\pi\)
\(48\) 0 0
\(49\) −3.89692 −0.556702
\(50\) 0 0
\(51\) 1.72928 0.242148
\(52\) 0 0
\(53\) −5.87859 −0.807487 −0.403744 − 0.914872i \(-0.632291\pi\)
−0.403744 + 0.914872i \(0.632291\pi\)
\(54\) 0 0
\(55\) −4.62620 −0.623796
\(56\) 0 0
\(57\) −4.62620 −0.612755
\(58\) 0 0
\(59\) 2.65847 0.346104 0.173052 − 0.984913i \(-0.444637\pi\)
0.173052 + 0.984913i \(0.444637\pi\)
\(60\) 0 0
\(61\) −3.72928 −0.477486 −0.238743 − 0.971083i \(-0.576735\pi\)
−0.238743 + 0.971083i \(0.576735\pi\)
\(62\) 0 0
\(63\) 1.76156 0.221935
\(64\) 0 0
\(65\) 4.38776 0.544234
\(66\) 0 0
\(67\) 7.64015 0.933393 0.466697 − 0.884418i \(-0.345444\pi\)
0.466697 + 0.884418i \(0.345444\pi\)
\(68\) 0 0
\(69\) −4.89692 −0.589519
\(70\) 0 0
\(71\) 3.69701 0.438754 0.219377 − 0.975640i \(-0.429598\pi\)
0.219377 + 0.975640i \(0.429598\pi\)
\(72\) 0 0
\(73\) 12.5371 1.46735 0.733676 − 0.679499i \(-0.237803\pi\)
0.733676 + 0.679499i \(0.237803\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −8.14931 −0.928700
\(78\) 0 0
\(79\) 3.87859 0.436376 0.218188 − 0.975907i \(-0.429985\pi\)
0.218188 + 0.975907i \(0.429985\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.747604 −0.0820602 −0.0410301 − 0.999158i \(-0.513064\pi\)
−0.0410301 + 0.999158i \(0.513064\pi\)
\(84\) 0 0
\(85\) 1.72928 0.187567
\(86\) 0 0
\(87\) −6.38776 −0.684839
\(88\) 0 0
\(89\) 3.49084 0.370028 0.185014 − 0.982736i \(-0.440767\pi\)
0.185014 + 0.982736i \(0.440767\pi\)
\(90\) 0 0
\(91\) 7.72928 0.810249
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −4.62620 −0.474638
\(96\) 0 0
\(97\) 13.0462 1.32464 0.662322 − 0.749219i \(-0.269571\pi\)
0.662322 + 0.749219i \(0.269571\pi\)
\(98\) 0 0
\(99\) −4.62620 −0.464950
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.1.3 3
4.3 odd 2 3720.2.a.m.1.1 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.m.1.1 ✓ 3 4.3 odd 2
7440.2.a.bw.1.3 3 1.1 even 1 trivial