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Properties

Label 7360.2.a.bl.1.1
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
Analytic rank $0$
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] = [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 = [2,0,-1,0,2,0,1,0,3,0,-4] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 \(2.56155\) 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)
 
\(f(q)\) \(=\) \(q-2.56155 q^{3} +1.00000 q^{5} -1.56155 q^{7} +3.56155 q^{9} -2.00000 q^{11} -0.561553 q^{13} -2.56155 q^{15} +5.56155 q^{17} -2.00000 q^{19} +4.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.43845 q^{27} -0.123106 q^{29} +8.12311 q^{31} +5.12311 q^{33} -1.56155 q^{35} +3.56155 q^{37} +1.43845 q^{39} -4.12311 q^{41} -10.2462 q^{43} +3.56155 q^{45} -3.68466 q^{47} -4.56155 q^{49} -14.2462 q^{51} -4.43845 q^{53} -2.00000 q^{55} +5.12311 q^{57} -5.56155 q^{59} +9.12311 q^{61} -5.56155 q^{63} -0.561553 q^{65} -11.5616 q^{67} -2.56155 q^{69} +5.00000 q^{71} +3.43845 q^{73} -2.56155 q^{75} +3.12311 q^{77} +9.12311 q^{79} -7.00000 q^{81} -4.68466 q^{83} +5.56155 q^{85} +0.315342 q^{87} +8.00000 q^{89} +0.876894 q^{91} -20.8078 q^{93} -2.00000 q^{95} -3.12311 q^{97} -7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 4 q^{11} + 3 q^{13} - q^{15} + 7 q^{17} - 4 q^{19} + 8 q^{21} + 2 q^{23} + 2 q^{25} - 7 q^{27} + 8 q^{29} + 8 q^{31} + 2 q^{33} + q^{35} + 3 q^{37} + 7 q^{39}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) −2.56155 −1.47891 −0.739457 − 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.56155 −0.590211 −0.295106 − 0.955465i \(-0.595355\pi\)
−0.295106 + 0.955465i \(0.595355\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −0.561553 −0.155747 −0.0778734 − 0.996963i \(-0.524813\pi\)
−0.0778734 + 0.996963i \(0.524813\pi\)
\(14\) 0 0
\(15\) −2.56155 −0.661390
\(16\) 0 0
\(17\) 5.56155 1.34887 0.674437 − 0.738332i \(-0.264386\pi\)
0.674437 + 0.738332i \(0.264386\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) −0.123106 −0.0228601 −0.0114301 − 0.999935i \(-0.503638\pi\)
−0.0114301 + 0.999935i \(0.503638\pi\)
\(30\) 0 0
\(31\) 8.12311 1.45895 0.729476 − 0.684006i \(-0.239764\pi\)
0.729476 + 0.684006i \(0.239764\pi\)
\(32\) 0 0
\(33\) 5.12311 0.891818
\(34\) 0 0
\(35\) −1.56155 −0.263951
\(36\) 0 0
\(37\) 3.56155 0.585516 0.292758 − 0.956187i \(-0.405427\pi\)
0.292758 + 0.956187i \(0.405427\pi\)
\(38\) 0 0
\(39\) 1.43845 0.230336
\(40\) 0 0
\(41\) −4.12311 −0.643921 −0.321960 − 0.946753i \(-0.604342\pi\)
−0.321960 + 0.946753i \(0.604342\pi\)
\(42\) 0 0
\(43\) −10.2462 −1.56253 −0.781266 − 0.624198i \(-0.785426\pi\)
−0.781266 + 0.624198i \(0.785426\pi\)
\(44\) 0 0
\(45\) 3.56155 0.530925
\(46\) 0 0
\(47\) −3.68466 −0.537463 −0.268731 − 0.963215i \(-0.586604\pi\)
−0.268731 + 0.963215i \(0.586604\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) −14.2462 −1.99487
\(52\) 0 0
\(53\) −4.43845 −0.609668 −0.304834 − 0.952406i \(-0.598601\pi\)
−0.304834 + 0.952406i \(0.598601\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 5.12311 0.678572
\(58\) 0 0
\(59\) −5.56155 −0.724053 −0.362026 − 0.932168i \(-0.617915\pi\)
−0.362026 + 0.932168i \(0.617915\pi\)
\(60\) 0 0
\(61\) 9.12311 1.16809 0.584047 − 0.811720i \(-0.301468\pi\)
0.584047 + 0.811720i \(0.301468\pi\)
\(62\) 0 0
\(63\) −5.56155 −0.700690
\(64\) 0 0
\(65\) −0.561553 −0.0696521
\(66\) 0 0
\(67\) −11.5616 −1.41247 −0.706234 − 0.707978i \(-0.749607\pi\)
−0.706234 + 0.707978i \(0.749607\pi\)
\(68\) 0 0
\(69\) −2.56155 −0.308375
\(70\) 0 0
\(71\) 5.00000 0.593391 0.296695 − 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) 0 0
\(73\) 3.43845 0.402440 0.201220 − 0.979546i \(-0.435509\pi\)
0.201220 + 0.979546i \(0.435509\pi\)
\(74\) 0 0
\(75\) −2.56155 −0.295783
\(76\) 0 0
\(77\) 3.12311 0.355911
\(78\) 0 0
\(79\) 9.12311 1.02643 0.513215 − 0.858260i \(-0.328454\pi\)
0.513215 + 0.858260i \(0.328454\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −4.68466 −0.514208 −0.257104 − 0.966384i \(-0.582768\pi\)
−0.257104 + 0.966384i \(0.582768\pi\)
\(84\) 0 0
\(85\) 5.56155 0.603235
\(86\) 0 0
\(87\) 0.315342 0.0338082
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0.876894 0.0919235
\(92\) 0 0
\(93\) −20.8078 −2.15766
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −3.12311 −0.317103 −0.158552 − 0.987351i \(-0.550682\pi\)
−0.158552 + 0.987351i \(0.550682\pi\)
\(98\) 0 0
\(99\) −7.12311 −0.715899
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.bl.1.1 2
4.3 odd 2 7360.2.a.bp.1.2 2
8.3 odd 2 920.2.a.e.1.1 ✓ 2
8.5 even 2 1840.2.a.o.1.2 2
24.11 even 2 8280.2.a.bf.1.2 2
40.3 even 4 4600.2.e.n.4049.1 4
40.19 odd 2 4600.2.a.t.1.2 2
40.27 even 4 4600.2.e.n.4049.4 4
40.29 even 2 9200.2.a.bq.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.e.1.1 ✓ 2 8.3 odd 2
1840.2.a.o.1.2 2 8.5 even 2
4600.2.a.t.1.2 2 40.19 odd 2
4600.2.e.n.4049.1 4 40.3 even 4
4600.2.e.n.4049.4 4 40.27 even 4
7360.2.a.bl.1.1 2 1.1 even 1 trivial
7360.2.a.bp.1.2 2 4.3 odd 2
8280.2.a.bf.1.2 2 24.11 even 2
9200.2.a.bq.1.1 2 40.29 even 2