Properties

Label 3720.2.a.r.1.1
Level $3720$
Weight $2$
Character 3720.1
Self dual yes
Analytic conductor $29.704$
Analytic rank $0$
Dimension $4$
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] = [3720,2,Mod(1,3720)] 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("3720.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3720.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,-4,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.7043495519\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.78292.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.78678\) of defining polynomial
Character \(\chi\) \(=\) 3720.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -2.78678 q^{7} +1.00000 q^{9} -4.92191 q^{11} -3.39720 q^{13} +1.00000 q^{15} -2.41782 q^{17} -6.92191 q^{19} +2.78678 q^{21} -3.80740 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.02062 q^{29} +1.00000 q^{31} +4.92191 q^{33} +2.78678 q^{35} +1.39720 q^{37} +3.39720 q^{39} +4.00000 q^{41} -1.38959 q^{43} -1.00000 q^{45} -2.37657 q^{47} +0.766162 q^{49} +2.41782 q^{51} +0.233838 q^{53} +4.92191 q^{55} +6.92191 q^{57} +3.29088 q^{59} -4.31150 q^{61} -2.78678 q^{63} +3.39720 q^{65} +0.176371 q^{67} +3.80740 q^{69} +3.58979 q^{71} +2.74554 q^{73} -1.00000 q^{75} +13.7163 q^{77} +5.06947 q^{79} +1.00000 q^{81} -5.63864 q^{83} +2.41782 q^{85} -1.02062 q^{87} +9.94253 q^{89} +9.46725 q^{91} -1.00000 q^{93} +6.92191 q^{95} -12.3680 q^{97} -4.92191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 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
\(7\) −2.78678 −1.05331 −0.526653 − 0.850081i \(-0.676553\pi\)
−0.526653 + 0.850081i \(0.676553\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.92191 −1.48401 −0.742006 − 0.670393i \(-0.766126\pi\)
−0.742006 + 0.670393i \(0.766126\pi\)
\(12\) 0 0
\(13\) −3.39720 −0.942213 −0.471106 − 0.882076i \(-0.656145\pi\)
−0.471106 + 0.882076i \(0.656145\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.41782 −0.586407 −0.293203 − 0.956050i \(-0.594721\pi\)
−0.293203 + 0.956050i \(0.594721\pi\)
\(18\) 0 0
\(19\) −6.92191 −1.58800 −0.793998 − 0.607921i \(-0.792004\pi\)
−0.793998 + 0.607921i \(0.792004\pi\)
\(20\) 0 0
\(21\) 2.78678 0.608126
\(22\) 0 0
\(23\) −3.80740 −0.793899 −0.396949 − 0.917841i \(-0.629931\pi\)
−0.396949 + 0.917841i \(0.629931\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.02062 0.189525 0.0947623 − 0.995500i \(-0.469791\pi\)
0.0947623 + 0.995500i \(0.469791\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) 4.92191 0.856795
\(34\) 0 0
\(35\) 2.78678 0.471052
\(36\) 0 0
\(37\) 1.39720 0.229698 0.114849 − 0.993383i \(-0.463362\pi\)
0.114849 + 0.993383i \(0.463362\pi\)
\(38\) 0 0
\(39\) 3.39720 0.543987
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −1.38959 −0.211910 −0.105955 − 0.994371i \(-0.533790\pi\)
−0.105955 + 0.994371i \(0.533790\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −2.37657 −0.346659 −0.173330 − 0.984864i \(-0.555453\pi\)
−0.173330 + 0.984864i \(0.555453\pi\)
\(48\) 0 0
\(49\) 0.766162 0.109452
\(50\) 0 0
\(51\) 2.41782 0.338562
\(52\) 0 0
\(53\) 0.233838 0.0321201 0.0160600 − 0.999871i \(-0.494888\pi\)
0.0160600 + 0.999871i \(0.494888\pi\)
\(54\) 0 0
\(55\) 4.92191 0.663671
\(56\) 0 0
\(57\) 6.92191 0.916830
\(58\) 0 0
\(59\) 3.29088 0.428436 0.214218 − 0.976786i \(-0.431280\pi\)
0.214218 + 0.976786i \(0.431280\pi\)
\(60\) 0 0
\(61\) −4.31150 −0.552031 −0.276015 − 0.961153i \(-0.589014\pi\)
−0.276015 + 0.961153i \(0.589014\pi\)
\(62\) 0 0
\(63\) −2.78678 −0.351102
\(64\) 0 0
\(65\) 3.39720 0.421370
\(66\) 0 0
\(67\) 0.176371 0.0215472 0.0107736 − 0.999942i \(-0.496571\pi\)
0.0107736 + 0.999942i \(0.496571\pi\)
\(68\) 0 0
\(69\) 3.80740 0.458358
\(70\) 0 0
\(71\) 3.58979 0.426030 0.213015 − 0.977049i \(-0.431672\pi\)
0.213015 + 0.977049i \(0.431672\pi\)
\(72\) 0 0
\(73\) 2.74554 0.321341 0.160671 − 0.987008i \(-0.448634\pi\)
0.160671 + 0.987008i \(0.448634\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 13.7163 1.56312
\(78\) 0 0
\(79\) 5.06947 0.570360 0.285180 − 0.958474i \(-0.407947\pi\)
0.285180 + 0.958474i \(0.407947\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.63864 −0.618921 −0.309461 − 0.950912i \(-0.600149\pi\)
−0.309461 + 0.950912i \(0.600149\pi\)
\(84\) 0 0
\(85\) 2.41782 0.262249
\(86\) 0 0
\(87\) −1.02062 −0.109422
\(88\) 0 0
\(89\) 9.94253 1.05391 0.526953 − 0.849894i \(-0.323334\pi\)
0.526953 + 0.849894i \(0.323334\pi\)
\(90\) 0 0
\(91\) 9.46725 0.992437
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 6.92191 0.710173
\(96\) 0 0
\(97\) −12.3680 −1.25578 −0.627888 − 0.778304i \(-0.716080\pi\)
−0.627888 + 0.778304i \(0.716080\pi\)
\(98\) 0 0
\(99\) −4.92191 −0.494671
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 3720.2.a.r.1.1 ✓ 4
4.3 odd 2 7440.2.a.bx.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.r.1.1 ✓ 4 1.1 even 1 trivial
7440.2.a.bx.1.4 4 4.3 odd 2