↑

Properties

Label 7360.2.a.cf.1.2
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.2
Root \(-1.86081\) 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)
 
\(f(q)\) \(=\) \(q+1.46260 q^{3} -1.00000 q^{5} -1.53740 q^{7} -0.860806 q^{9} -0.860806 q^{11} -0.139194 q^{13} -1.46260 q^{15} +5.50761 q^{17} +5.25901 q^{19} -2.24860 q^{21} +1.00000 q^{23} +1.00000 q^{25} -5.64681 q^{27} -9.76663 q^{29} -6.78600 q^{31} -1.25901 q^{33} +1.53740 q^{35} +12.0900 q^{37} -0.203585 q^{39} -9.98062 q^{41} +11.4432 q^{43} +0.860806 q^{45} +2.32340 q^{47} -4.63640 q^{49} +8.05543 q^{51} -0.149606 q^{53} +0.860806 q^{55} +7.69182 q^{57} -11.0152 q^{59} -4.43281 q^{61} +1.32340 q^{63} +0.139194 q^{65} -10.4972 q^{67} +1.46260 q^{69} -7.31299 q^{71} +7.11982 q^{73} +1.46260 q^{75} +1.32340 q^{77} -6.79641 q^{79} -5.67660 q^{81} +10.6468 q^{83} -5.50761 q^{85} -14.2847 q^{87} -17.2936 q^{89} +0.213997 q^{91} -9.92520 q^{93} -5.25901 q^{95} +1.93561 q^{97} +0.740987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 3 q^{5} - 7 q^{7} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 2 q^{15} - 5 q^{17} + 7 q^{19} + 6 q^{21} + 3 q^{23} + 3 q^{25} - q^{27} + q^{29} - 10 q^{31} + 5 q^{33} + 7 q^{35} - 2 q^{37} - 7 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) \) 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
\(3\) 1.46260 0.844432 0.422216 − 0.906495i \(-0.361252\pi\)
0.422216 + 0.906495i \(0.361252\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.53740 −0.581083 −0.290542 − 0.956862i \(-0.593835\pi\)
−0.290542 + 0.956862i \(0.593835\pi\)
\(8\) 0 0
\(9\) −0.860806 −0.286935
\(10\) 0 0
\(11\) −0.860806 −0.259543 −0.129771 − 0.991544i \(-0.541424\pi\)
−0.129771 + 0.991544i \(0.541424\pi\)
\(12\) 0 0
\(13\) −0.139194 −0.0386055 −0.0193028 − 0.999814i \(-0.506145\pi\)
−0.0193028 + 0.999814i \(0.506145\pi\)
\(14\) 0 0
\(15\) −1.46260 −0.377641
\(16\) 0 0
\(17\) 5.50761 1.33579 0.667896 − 0.744254i \(-0.267195\pi\)
0.667896 + 0.744254i \(0.267195\pi\)
\(18\) 0 0
\(19\) 5.25901 1.20650 0.603250 − 0.797552i \(-0.293872\pi\)
0.603250 + 0.797552i \(0.293872\pi\)
\(20\) 0 0
\(21\) −2.24860 −0.490685
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.64681 −1.08673
\(28\) 0 0
\(29\) −9.76663 −1.81362 −0.906809 − 0.421543i \(-0.861489\pi\)
−0.906809 + 0.421543i \(0.861489\pi\)
\(30\) 0 0
\(31\) −6.78600 −1.21880 −0.609401 − 0.792862i \(-0.708590\pi\)
−0.609401 + 0.792862i \(0.708590\pi\)
\(32\) 0 0
\(33\) −1.25901 −0.219166
\(34\) 0 0
\(35\) 1.53740 0.259868
\(36\) 0 0
\(37\) 12.0900 1.98759 0.993795 − 0.111232i \(-0.0354796\pi\)
0.993795 + 0.111232i \(0.0354796\pi\)
\(38\) 0 0
\(39\) −0.203585 −0.0325997
\(40\) 0 0
\(41\) −9.98062 −1.55871 −0.779356 − 0.626582i \(-0.784454\pi\)
−0.779356 + 0.626582i \(0.784454\pi\)
\(42\) 0 0
\(43\) 11.4432 1.74508 0.872538 − 0.488547i \(-0.162473\pi\)
0.872538 + 0.488547i \(0.162473\pi\)
\(44\) 0 0
\(45\) 0.860806 0.128321
\(46\) 0 0
\(47\) 2.32340 0.338903 0.169452 − 0.985538i \(-0.445800\pi\)
0.169452 + 0.985538i \(0.445800\pi\)
\(48\) 0 0
\(49\) −4.63640 −0.662342
\(50\) 0 0
\(51\) 8.05543 1.12799
\(52\) 0 0
\(53\) −0.149606 −0.0205500 −0.0102750 − 0.999947i \(-0.503271\pi\)
−0.0102750 + 0.999947i \(0.503271\pi\)
\(54\) 0 0
\(55\) 0.860806 0.116071
\(56\) 0 0
\(57\) 7.69182 1.01881
\(58\) 0 0
\(59\) −11.0152 −1.43406 −0.717030 − 0.697042i \(-0.754499\pi\)
−0.717030 + 0.697042i \(0.754499\pi\)
\(60\) 0 0
\(61\) −4.43281 −0.567563 −0.283782 − 0.958889i \(-0.591589\pi\)
−0.283782 + 0.958889i \(0.591589\pi\)
\(62\) 0 0
\(63\) 1.32340 0.166733
\(64\) 0 0
\(65\) 0.139194 0.0172649
\(66\) 0 0
\(67\) −10.4972 −1.28244 −0.641219 − 0.767358i \(-0.721571\pi\)
−0.641219 + 0.767358i \(0.721571\pi\)
\(68\) 0 0
\(69\) 1.46260 0.176076
\(70\) 0 0
\(71\) −7.31299 −0.867892 −0.433946 − 0.900939i \(-0.642879\pi\)
−0.433946 + 0.900939i \(0.642879\pi\)
\(72\) 0 0
\(73\) 7.11982 0.833312 0.416656 − 0.909064i \(-0.363202\pi\)
0.416656 + 0.909064i \(0.363202\pi\)
\(74\) 0 0
\(75\) 1.46260 0.168886
\(76\) 0 0
\(77\) 1.32340 0.150816
\(78\) 0 0
\(79\) −6.79641 −0.764656 −0.382328 − 0.924027i \(-0.624878\pi\)
−0.382328 + 0.924027i \(0.624878\pi\)
\(80\) 0 0
\(81\) −5.67660 −0.630733
\(82\) 0 0
\(83\) 10.6468 1.16864 0.584320 − 0.811524i \(-0.301361\pi\)
0.584320 + 0.811524i \(0.301361\pi\)
\(84\) 0 0
\(85\) −5.50761 −0.597385
\(86\) 0 0
\(87\) −14.2847 −1.53148
\(88\) 0 0
\(89\) −17.2936 −1.83312 −0.916560 − 0.399897i \(-0.869046\pi\)
−0.916560 + 0.399897i \(0.869046\pi\)
\(90\) 0 0
\(91\) 0.213997 0.0224330
\(92\) 0 0
\(93\) −9.92520 −1.02919
\(94\) 0 0
\(95\) −5.25901 −0.539563
\(96\) 0 0
\(97\) 1.93561 0.196531 0.0982657 − 0.995160i \(-0.468671\pi\)
0.0982657 + 0.995160i \(0.468671\pi\)
\(98\) 0 0
\(99\) 0.740987 0.0744720
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.1.2 3
4.3 odd 2 7360.2.a.bw.1.2 3
8.3 odd 2 920.2.a.i.1.2 ✓ 3
8.5 even 2 1840.2.a.q.1.2 3
24.11 even 2 8280.2.a.bl.1.2 3
40.3 even 4 4600.2.e.q.4049.4 6
40.19 odd 2 4600.2.a.v.1.2 3
40.27 even 4 4600.2.e.q.4049.3 6
40.29 even 2 9200.2.a.ci.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.2 ✓ 3 8.3 odd 2
1840.2.a.q.1.2 3 8.5 even 2
4600.2.a.v.1.2 3 40.19 odd 2
4600.2.e.q.4049.3 6 40.27 even 4
4600.2.e.q.4049.4 6 40.3 even 4
7360.2.a.bw.1.2 3 4.3 odd 2
7360.2.a.cf.1.2 3 1.1 even 1 trivial
8280.2.a.bl.1.2 3 24.11 even 2
9200.2.a.ci.1.2 3 40.29 even 2