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Properties

Label 7360.2.a.cf.1.1
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
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] = [7360,2,Mod(1,7360)] 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("7360.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,2,0,-3,0,-7,0,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(58.7698958877\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 7360.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.93543 q^{3} -1.00000 q^{5} -4.93543 q^{7} +0.745898 q^{9} +0.745898 q^{11} -1.74590 q^{13} +1.93543 q^{15} -6.10856 q^{17} +5.44364 q^{19} +9.55220 q^{21} +1.00000 q^{23} +1.00000 q^{25} +4.36266 q^{27} +1.66492 q^{29} +1.61676 q^{31} -1.44364 q^{33} +4.93543 q^{35} -4.34625 q^{37} +3.37907 q^{39} -6.95184 q^{41} +5.01641 q^{43} -0.745898 q^{45} -2.68133 q^{47} +17.3585 q^{49} +11.8227 q^{51} -13.7417 q^{53} -0.745898 q^{55} -10.5358 q^{57} +12.2171 q^{59} +13.9794 q^{61} -3.68133 q^{63} +1.74590 q^{65} +13.1044 q^{67} -1.93543 q^{69} +9.67716 q^{71} +5.69774 q^{73} -1.93543 q^{75} -3.68133 q^{77} -10.3791 q^{79} -10.6813 q^{81} +0.637339 q^{83} +6.10856 q^{85} -3.22235 q^{87} +2.72532 q^{89} +8.61676 q^{91} -3.12914 q^{93} -5.44364 q^{95} +7.12497 q^{97} +0.556364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 3 q^{5} - 7 q^{7} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 2 q^{15} - 5 q^{17} + 7 q^{19} + 6 q^{21} + 3 q^{23} + 3 q^{25} - q^{27} + q^{29} - 10 q^{31} + 5 q^{33} + 7 q^{35} - 2 q^{37} - 7 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) \) 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
\(3\) −1.93543 −1.11742 −0.558711 − 0.829362i \(-0.688704\pi\)
−0.558711 + 0.829362i \(0.688704\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.93543 −1.86542 −0.932709 − 0.360630i \(-0.882562\pi\)
−0.932709 + 0.360630i \(0.882562\pi\)
\(8\) 0 0
\(9\) 0.745898 0.248633
\(10\) 0 0
\(11\) 0.745898 0.224897 0.112448 − 0.993658i \(-0.464131\pi\)
0.112448 + 0.993658i \(0.464131\pi\)
\(12\) 0 0
\(13\) −1.74590 −0.484225 −0.242113 − 0.970248i \(-0.577840\pi\)
−0.242113 + 0.970248i \(0.577840\pi\)
\(14\) 0 0
\(15\) 1.93543 0.499726
\(16\) 0 0
\(17\) −6.10856 −1.48154 −0.740772 − 0.671757i \(-0.765540\pi\)
−0.740772 + 0.671757i \(0.765540\pi\)
\(18\) 0 0
\(19\) 5.44364 1.24886 0.624428 − 0.781083i \(-0.285332\pi\)
0.624428 + 0.781083i \(0.285332\pi\)
\(20\) 0 0
\(21\) 9.55220 2.08446
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.36266 0.839595
\(28\) 0 0
\(29\) 1.66492 0.309169 0.154584 − 0.987980i \(-0.450596\pi\)
0.154584 + 0.987980i \(0.450596\pi\)
\(30\) 0 0
\(31\) 1.61676 0.290379 0.145190 − 0.989404i \(-0.453621\pi\)
0.145190 + 0.989404i \(0.453621\pi\)
\(32\) 0 0
\(33\) −1.44364 −0.251305
\(34\) 0 0
\(35\) 4.93543 0.834240
\(36\) 0 0
\(37\) −4.34625 −0.714520 −0.357260 − 0.934005i \(-0.616289\pi\)
−0.357260 + 0.934005i \(0.616289\pi\)
\(38\) 0 0
\(39\) 3.37907 0.541084
\(40\) 0 0
\(41\) −6.95184 −1.08569 −0.542847 − 0.839831i \(-0.682654\pi\)
−0.542847 + 0.839831i \(0.682654\pi\)
\(42\) 0 0
\(43\) 5.01641 0.764995 0.382497 − 0.923957i \(-0.375064\pi\)
0.382497 + 0.923957i \(0.375064\pi\)
\(44\) 0 0
\(45\) −0.745898 −0.111192
\(46\) 0 0
\(47\) −2.68133 −0.391112 −0.195556 − 0.980693i \(-0.562651\pi\)
−0.195556 + 0.980693i \(0.562651\pi\)
\(48\) 0 0
\(49\) 17.3585 2.47978
\(50\) 0 0
\(51\) 11.8227 1.65551
\(52\) 0 0
\(53\) −13.7417 −1.88757 −0.943786 − 0.330558i \(-0.892763\pi\)
−0.943786 + 0.330558i \(0.892763\pi\)
\(54\) 0 0
\(55\) −0.745898 −0.100577
\(56\) 0 0
\(57\) −10.5358 −1.39550
\(58\) 0 0
\(59\) 12.2171 1.59053 0.795267 − 0.606260i \(-0.207331\pi\)
0.795267 + 0.606260i \(0.207331\pi\)
\(60\) 0 0
\(61\) 13.9794 1.78988 0.894941 − 0.446185i \(-0.147218\pi\)
0.894941 + 0.446185i \(0.147218\pi\)
\(62\) 0 0
\(63\) −3.68133 −0.463804
\(64\) 0 0
\(65\) 1.74590 0.216552
\(66\) 0 0
\(67\) 13.1044 1.60096 0.800478 − 0.599362i \(-0.204579\pi\)
0.800478 + 0.599362i \(0.204579\pi\)
\(68\) 0 0
\(69\) −1.93543 −0.232999
\(70\) 0 0
\(71\) 9.67716 1.14847 0.574234 − 0.818691i \(-0.305300\pi\)
0.574234 + 0.818691i \(0.305300\pi\)
\(72\) 0 0
\(73\) 5.69774 0.666870 0.333435 − 0.942773i \(-0.391792\pi\)
0.333435 + 0.942773i \(0.391792\pi\)
\(74\) 0 0
\(75\) −1.93543 −0.223484
\(76\) 0 0
\(77\) −3.68133 −0.419527
\(78\) 0 0
\(79\) −10.3791 −1.16774 −0.583868 − 0.811848i \(-0.698462\pi\)
−0.583868 + 0.811848i \(0.698462\pi\)
\(80\) 0 0
\(81\) −10.6813 −1.18681
\(82\) 0 0
\(83\) 0.637339 0.0699570 0.0349785 − 0.999388i \(-0.488864\pi\)
0.0349785 + 0.999388i \(0.488864\pi\)
\(84\) 0 0
\(85\) 6.10856 0.662566
\(86\) 0 0
\(87\) −3.22235 −0.345472
\(88\) 0 0
\(89\) 2.72532 0.288884 0.144442 − 0.989513i \(-0.453861\pi\)
0.144442 + 0.989513i \(0.453861\pi\)
\(90\) 0 0
\(91\) 8.61676 0.903282
\(92\) 0 0
\(93\) −3.12914 −0.324476
\(94\) 0 0
\(95\) −5.44364 −0.558505
\(96\) 0 0
\(97\) 7.12497 0.723431 0.361715 − 0.932289i \(-0.382191\pi\)
0.361715 + 0.932289i \(0.382191\pi\)
\(98\) 0 0
\(99\) 0.556364 0.0559167
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 7360.2.a.cf.1.1 3
4.3 odd 2 7360.2.a.bw.1.3 3
8.3 odd 2 920.2.a.i.1.1 ✓ 3
8.5 even 2 1840.2.a.q.1.3 3
24.11 even 2 8280.2.a.bl.1.3 3
40.3 even 4 4600.2.e.q.4049.2 6
40.19 odd 2 4600.2.a.v.1.3 3
40.27 even 4 4600.2.e.q.4049.5 6
40.29 even 2 9200.2.a.ci.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.1 ✓ 3 8.3 odd 2
1840.2.a.q.1.3 3 8.5 even 2
4600.2.a.v.1.3 3 40.19 odd 2
4600.2.e.q.4049.2 6 40.3 even 4
4600.2.e.q.4049.5 6 40.27 even 4
7360.2.a.bw.1.3 3 4.3 odd 2
7360.2.a.cf.1.1 3 1.1 even 1 trivial
8280.2.a.bl.1.3 3 24.11 even 2
9200.2.a.ci.1.1 3 40.29 even 2