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Properties

Label 7360.2.a.bl.1.2
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 1.2
Root \(-1.56155\) 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)
 
\(f(q)\) \(=\) \(q+1.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} -0.561553 q^{9} -2.00000 q^{11} +3.56155 q^{13} +1.56155 q^{15} +1.43845 q^{17} -2.00000 q^{19} +4.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -5.56155 q^{27} +8.12311 q^{29} -0.123106 q^{31} -3.12311 q^{33} +2.56155 q^{35} -0.561553 q^{37} +5.56155 q^{39} +4.12311 q^{41} +6.24621 q^{43} -0.561553 q^{45} +8.68466 q^{47} -0.438447 q^{49} +2.24621 q^{51} -8.56155 q^{53} -2.00000 q^{55} -3.12311 q^{57} -1.43845 q^{59} +0.876894 q^{61} -1.43845 q^{63} +3.56155 q^{65} -7.43845 q^{67} +1.56155 q^{69} +5.00000 q^{71} +7.56155 q^{73} +1.56155 q^{75} -5.12311 q^{77} +0.876894 q^{79} -7.00000 q^{81} +7.68466 q^{83} +1.43845 q^{85} +12.6847 q^{87} +8.00000 q^{89} +9.12311 q^{91} -0.192236 q^{93} -2.00000 q^{95} +5.12311 q^{97} +1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 4 q^{11} + 3 q^{13} - q^{15} + 7 q^{17} - 4 q^{19} + 8 q^{21} + 2 q^{23} + 2 q^{25} - 7 q^{27} + 8 q^{29} + 8 q^{31} + 2 q^{33} + q^{35} + 3 q^{37} + 7 q^{39}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) 1.56155 0.901563 0.450781 − 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.56155 0.968176 0.484088 − 0.875019i \(-0.339151\pi\)
0.484088 + 0.875019i \(0.339151\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 3.56155 0.987797 0.493899 − 0.869520i \(-0.335571\pi\)
0.493899 + 0.869520i \(0.335571\pi\)
\(14\) 0 0
\(15\) 1.56155 0.403191
\(16\) 0 0
\(17\) 1.43845 0.348875 0.174437 − 0.984668i \(-0.444189\pi\)
0.174437 + 0.984668i \(0.444189\pi\)
\(18\) 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
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) 8.12311 1.50842 0.754211 − 0.656632i \(-0.228019\pi\)
0.754211 + 0.656632i \(0.228019\pi\)
\(30\) 0 0
\(31\) −0.123106 −0.0221104 −0.0110552 − 0.999939i \(-0.503519\pi\)
−0.0110552 + 0.999939i \(0.503519\pi\)
\(32\) 0 0
\(33\) −3.12311 −0.543663
\(34\) 0 0
\(35\) 2.56155 0.432981
\(36\) 0 0
\(37\) −0.561553 −0.0923187 −0.0461594 − 0.998934i \(-0.514698\pi\)
−0.0461594 + 0.998934i \(0.514698\pi\)
\(38\) 0 0
\(39\) 5.56155 0.890561
\(40\) 0 0
\(41\) 4.12311 0.643921 0.321960 − 0.946753i \(-0.395658\pi\)
0.321960 + 0.946753i \(0.395658\pi\)
\(42\) 0 0
\(43\) 6.24621 0.952538 0.476269 − 0.879300i \(-0.341989\pi\)
0.476269 + 0.879300i \(0.341989\pi\)
\(44\) 0 0
\(45\) −0.561553 −0.0837114
\(46\) 0 0
\(47\) 8.68466 1.26679 0.633394 − 0.773830i \(-0.281661\pi\)
0.633394 + 0.773830i \(0.281661\pi\)
\(48\) 0 0
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) 2.24621 0.314532
\(52\) 0 0
\(53\) −8.56155 −1.17602 −0.588010 − 0.808854i \(-0.700088\pi\)
−0.588010 + 0.808854i \(0.700088\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −3.12311 −0.413665
\(58\) 0 0
\(59\) −1.43845 −0.187270 −0.0936349 − 0.995607i \(-0.529849\pi\)
−0.0936349 + 0.995607i \(0.529849\pi\)
\(60\) 0 0
\(61\) 0.876894 0.112275 0.0561374 − 0.998423i \(-0.482122\pi\)
0.0561374 + 0.998423i \(0.482122\pi\)
\(62\) 0 0
\(63\) −1.43845 −0.181227
\(64\) 0 0
\(65\) 3.56155 0.441756
\(66\) 0 0
\(67\) −7.43845 −0.908751 −0.454375 − 0.890810i \(-0.650138\pi\)
−0.454375 + 0.890810i \(0.650138\pi\)
\(68\) 0 0
\(69\) 1.56155 0.187989
\(70\) 0 0
\(71\) 5.00000 0.593391 0.296695 − 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) 0 0
\(73\) 7.56155 0.885013 0.442506 − 0.896765i \(-0.354089\pi\)
0.442506 + 0.896765i \(0.354089\pi\)
\(74\) 0 0
\(75\) 1.56155 0.180313
\(76\) 0 0
\(77\) −5.12311 −0.583832
\(78\) 0 0
\(79\) 0.876894 0.0986583 0.0493292 − 0.998783i \(-0.484292\pi\)
0.0493292 + 0.998783i \(0.484292\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 7.68466 0.843501 0.421750 − 0.906712i \(-0.361416\pi\)
0.421750 + 0.906712i \(0.361416\pi\)
\(84\) 0 0
\(85\) 1.43845 0.156022
\(86\) 0 0
\(87\) 12.6847 1.35994
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 9.12311 0.956361
\(92\) 0 0
\(93\) −0.192236 −0.0199339
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 5.12311 0.520173 0.260086 − 0.965585i \(-0.416249\pi\)
0.260086 + 0.965585i \(0.416249\pi\)
\(98\) 0 0
\(99\) 1.12311 0.112876
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.1.2 2
4.3 odd 2 7360.2.a.bp.1.1 2
8.3 odd 2 920.2.a.e.1.2 ✓ 2
8.5 even 2 1840.2.a.o.1.1 2
24.11 even 2 8280.2.a.bf.1.1 2
40.3 even 4 4600.2.e.n.4049.3 4
40.19 odd 2 4600.2.a.t.1.1 2
40.27 even 4 4600.2.e.n.4049.2 4
40.29 even 2 9200.2.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.e.1.2 ✓ 2 8.3 odd 2
1840.2.a.o.1.1 2 8.5 even 2
4600.2.a.t.1.1 2 40.19 odd 2
4600.2.e.n.4049.2 4 40.27 even 4
4600.2.e.n.4049.3 4 40.3 even 4
7360.2.a.bl.1.2 2 1.1 even 1 trivial
7360.2.a.bp.1.1 2 4.3 odd 2
8280.2.a.bf.1.1 2 24.11 even 2
9200.2.a.bq.1.2 2 40.29 even 2