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Properties

Label 7440.2.a.bh.1.1
Level $7440$
Weight $2$
Character 7440.1
Self dual yes
Analytic conductor $59.409$
Analytic rank $1$
Dimension $2$
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 = [2,0,-2,0,2,0,4,0,2,0,0,0,-8,0,-2,0,0] 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: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(-1.41421\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +0.585786 q^{7} +1.00000 q^{9} +2.82843 q^{11} -5.41421 q^{13} -1.00000 q^{15} +2.82843 q^{17} +2.82843 q^{19} -0.585786 q^{21} -3.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} -10.2426 q^{29} +1.00000 q^{31} -2.82843 q^{33} +0.585786 q^{35} +7.07107 q^{37} +5.41421 q^{39} -4.82843 q^{41} -9.65685 q^{43} +1.00000 q^{45} +11.6569 q^{47} -6.65685 q^{49} -2.82843 q^{51} -5.17157 q^{53} +2.82843 q^{55} -2.82843 q^{57} -5.41421 q^{59} -4.82843 q^{61} +0.585786 q^{63} -5.41421 q^{65} +9.07107 q^{67} +3.65685 q^{69} +5.41421 q^{71} -9.41421 q^{73} -1.00000 q^{75} +1.65685 q^{77} -8.82843 q^{79} +1.00000 q^{81} +2.00000 q^{83} +2.82843 q^{85} +10.2426 q^{87} -14.7279 q^{89} -3.17157 q^{91} -1.00000 q^{93} +2.82843 q^{95} +3.65685 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} - 8 q^{13} - 2 q^{15} - 4 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 12 q^{29} + 2 q^{31} + 4 q^{35} + 8 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 12 q^{47} - 2 q^{49}+ \cdots - 4 q^{97}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.585786 0.221406 0.110703 − 0.993854i \(-0.464690\pi\)
0.110703 + 0.993854i \(0.464690\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 − 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) −5.41421 −1.50163 −0.750816 − 0.660511i \(-0.770340\pi\)
−0.750816 + 0.660511i \(0.770340\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 − 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 − 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) −0.585786 −0.127829
\(22\) 0 0
\(23\) −3.65685 −0.762507 −0.381253 − 0.924471i \(-0.624507\pi\)
−0.381253 + 0.924471i \(0.624507\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.2426 −1.90201 −0.951005 − 0.309175i \(-0.899947\pi\)
−0.951005 + 0.309175i \(0.899947\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) 0.585786 0.0990160
\(36\) 0 0
\(37\) 7.07107 1.16248 0.581238 − 0.813733i \(-0.302568\pi\)
0.581238 + 0.813733i \(0.302568\pi\)
\(38\) 0 0
\(39\) 5.41421 0.866968
\(40\) 0 0
\(41\) −4.82843 −0.754074 −0.377037 − 0.926198i \(-0.623057\pi\)
−0.377037 + 0.926198i \(0.623057\pi\)
\(42\) 0 0
\(43\) −9.65685 −1.47266 −0.736328 − 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 11.6569 1.70033 0.850163 − 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) 0 0
\(49\) −6.65685 −0.950979
\(50\) 0 0
\(51\) −2.82843 −0.396059
\(52\) 0 0
\(53\) −5.17157 −0.710370 −0.355185 − 0.934796i \(-0.615582\pi\)
−0.355185 + 0.934796i \(0.615582\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) 0 0
\(59\) −5.41421 −0.704871 −0.352435 − 0.935836i \(-0.614646\pi\)
−0.352435 + 0.935836i \(0.614646\pi\)
\(60\) 0 0
\(61\) −4.82843 −0.618217 −0.309108 − 0.951027i \(-0.600031\pi\)
−0.309108 + 0.951027i \(0.600031\pi\)
\(62\) 0 0
\(63\) 0.585786 0.0738022
\(64\) 0 0
\(65\) −5.41421 −0.671551
\(66\) 0 0
\(67\) 9.07107 1.10821 0.554104 − 0.832448i \(-0.313061\pi\)
0.554104 + 0.832448i \(0.313061\pi\)
\(68\) 0 0
\(69\) 3.65685 0.440234
\(70\) 0 0
\(71\) 5.41421 0.642549 0.321274 − 0.946986i \(-0.395889\pi\)
0.321274 + 0.946986i \(0.395889\pi\)
\(72\) 0 0
\(73\) −9.41421 −1.10185 −0.550925 − 0.834555i \(-0.685725\pi\)
−0.550925 + 0.834555i \(0.685725\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 1.65685 0.188816
\(78\) 0 0
\(79\) −8.82843 −0.993276 −0.496638 − 0.867958i \(-0.665432\pi\)
−0.496638 + 0.867958i \(0.665432\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 0 0
\(87\) 10.2426 1.09813
\(88\) 0 0
\(89\) −14.7279 −1.56116 −0.780578 − 0.625058i \(-0.785075\pi\)
−0.780578 + 0.625058i \(0.785075\pi\)
\(90\) 0 0
\(91\) −3.17157 −0.332471
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) 3.65685 0.371297 0.185649 − 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
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.bh.1.1 2
4.3 odd 2 3720.2.a.j.1.2 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.j.1.2 ✓ 2 4.3 odd 2
7440.2.a.bh.1.1 2 1.1 even 1 trivial