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Properties

Label 7360.2.a.bq.1.2
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
Analytic rank $1$
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 = [2,0,1,0,2,0,1,0,5,0,-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: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) 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 230)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.79129\) 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)
 
\(f(q)\) \(=\) \(q+2.79129 q^{3} +1.00000 q^{5} -1.79129 q^{7} +4.79129 q^{9} +0.791288 q^{11} -5.79129 q^{13} +2.79129 q^{15} +0.791288 q^{17} -5.79129 q^{19} -5.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -7.58258 q^{29} -3.37386 q^{31} +2.20871 q^{33} -1.79129 q^{35} +4.00000 q^{37} -16.1652 q^{39} -6.79129 q^{41} -11.1652 q^{43} +4.79129 q^{45} -4.41742 q^{47} -3.79129 q^{49} +2.20871 q^{51} -6.00000 q^{53} +0.791288 q^{55} -16.1652 q^{57} +13.5826 q^{59} -10.3739 q^{61} -8.58258 q^{63} -5.79129 q^{65} -11.1652 q^{67} +2.79129 q^{69} +8.37386 q^{71} +12.7477 q^{73} +2.79129 q^{75} -1.41742 q^{77} +8.00000 q^{79} -0.417424 q^{81} +6.00000 q^{83} +0.791288 q^{85} -21.1652 q^{87} +15.1652 q^{89} +10.3739 q^{91} -9.41742 q^{93} -5.79129 q^{95} -7.95644 q^{97} +3.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + q^{7} + 5 q^{9} - 3 q^{11} - 7 q^{13} + q^{15} - 3 q^{17} - 7 q^{19} - 10 q^{21} + 2 q^{23} + 2 q^{25} + 10 q^{27} - 6 q^{29} + 7 q^{31} + 9 q^{33} + q^{35} + 8 q^{37} - 14 q^{39}+ \cdots + 3 q^{99}+O(q^{100}) \) 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
\(3\) 2.79129 1.61155 0.805775 − 0.592221i \(-0.201749\pi\)
0.805775 + 0.592221i \(0.201749\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.79129 −0.677043 −0.338522 − 0.940959i \(-0.609927\pi\)
−0.338522 + 0.940959i \(0.609927\pi\)
\(8\) 0 0
\(9\) 4.79129 1.59710
\(10\) 0 0
\(11\) 0.791288 0.238582 0.119291 − 0.992859i \(-0.461938\pi\)
0.119291 + 0.992859i \(0.461938\pi\)
\(12\) 0 0
\(13\) −5.79129 −1.60621 −0.803107 − 0.595835i \(-0.796821\pi\)
−0.803107 + 0.595835i \(0.796821\pi\)
\(14\) 0 0
\(15\) 2.79129 0.720707
\(16\) 0 0
\(17\) 0.791288 0.191915 0.0959577 − 0.995385i \(-0.469409\pi\)
0.0959577 + 0.995385i \(0.469409\pi\)
\(18\) 0 0
\(19\) −5.79129 −1.32861 −0.664306 − 0.747460i \(-0.731273\pi\)
−0.664306 + 0.747460i \(0.731273\pi\)
\(20\) 0 0
\(21\) −5.00000 −1.09109
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −7.58258 −1.40805 −0.704024 − 0.710176i \(-0.748615\pi\)
−0.704024 + 0.710176i \(0.748615\pi\)
\(30\) 0 0
\(31\) −3.37386 −0.605964 −0.302982 − 0.952996i \(-0.597982\pi\)
−0.302982 + 0.952996i \(0.597982\pi\)
\(32\) 0 0
\(33\) 2.20871 0.384487
\(34\) 0 0
\(35\) −1.79129 −0.302783
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −16.1652 −2.58850
\(40\) 0 0
\(41\) −6.79129 −1.06062 −0.530310 − 0.847804i \(-0.677925\pi\)
−0.530310 + 0.847804i \(0.677925\pi\)
\(42\) 0 0
\(43\) −11.1652 −1.70267 −0.851335 − 0.524623i \(-0.824206\pi\)
−0.851335 + 0.524623i \(0.824206\pi\)
\(44\) 0 0
\(45\) 4.79129 0.714243
\(46\) 0 0
\(47\) −4.41742 −0.644348 −0.322174 − 0.946681i \(-0.604414\pi\)
−0.322174 + 0.946681i \(0.604414\pi\)
\(48\) 0 0
\(49\) −3.79129 −0.541613
\(50\) 0 0
\(51\) 2.20871 0.309282
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0.791288 0.106697
\(56\) 0 0
\(57\) −16.1652 −2.14113
\(58\) 0 0
\(59\) 13.5826 1.76830 0.884150 − 0.467202i \(-0.154738\pi\)
0.884150 + 0.467202i \(0.154738\pi\)
\(60\) 0 0
\(61\) −10.3739 −1.32824 −0.664119 − 0.747627i \(-0.731193\pi\)
−0.664119 + 0.747627i \(0.731193\pi\)
\(62\) 0 0
\(63\) −8.58258 −1.08130
\(64\) 0 0
\(65\) −5.79129 −0.718321
\(66\) 0 0
\(67\) −11.1652 −1.36404 −0.682020 − 0.731333i \(-0.738898\pi\)
−0.682020 + 0.731333i \(0.738898\pi\)
\(68\) 0 0
\(69\) 2.79129 0.336032
\(70\) 0 0
\(71\) 8.37386 0.993795 0.496897 − 0.867809i \(-0.334473\pi\)
0.496897 + 0.867809i \(0.334473\pi\)
\(72\) 0 0
\(73\) 12.7477 1.49201 0.746004 − 0.665941i \(-0.231970\pi\)
0.746004 + 0.665941i \(0.231970\pi\)
\(74\) 0 0
\(75\) 2.79129 0.322310
\(76\) 0 0
\(77\) −1.41742 −0.161530
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0.791288 0.0858272
\(86\) 0 0
\(87\) −21.1652 −2.26914
\(88\) 0 0
\(89\) 15.1652 1.60750 0.803751 − 0.594965i \(-0.202834\pi\)
0.803751 + 0.594965i \(0.202834\pi\)
\(90\) 0 0
\(91\) 10.3739 1.08748
\(92\) 0 0
\(93\) −9.41742 −0.976541
\(94\) 0 0
\(95\) −5.79129 −0.594174
\(96\) 0 0
\(97\) −7.95644 −0.807854 −0.403927 − 0.914791i \(-0.632355\pi\)
−0.403927 + 0.914791i \(0.632355\pi\)
\(98\) 0 0
\(99\) 3.79129 0.381039
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.bq.1.2 2
4.3 odd 2 7360.2.a.bk.1.1 2
8.3 odd 2 1840.2.a.n.1.2 2
8.5 even 2 230.2.a.a.1.1 ✓ 2
24.5 odd 2 2070.2.a.x.1.1 2
40.13 odd 4 1150.2.b.g.599.3 4
40.19 odd 2 9200.2.a.bs.1.1 2
40.29 even 2 1150.2.a.o.1.2 2
40.37 odd 4 1150.2.b.g.599.2 4
184.45 odd 2 5290.2.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.1 ✓ 2 8.5 even 2
1150.2.a.o.1.2 2 40.29 even 2
1150.2.b.g.599.2 4 40.37 odd 4
1150.2.b.g.599.3 4 40.13 odd 4
1840.2.a.n.1.2 2 8.3 odd 2
2070.2.a.x.1.1 2 24.5 odd 2
5290.2.a.e.1.1 2 184.45 odd 2
7360.2.a.bk.1.1 2 4.3 odd 2
7360.2.a.bq.1.2 2 1.1 even 1 trivial
9200.2.a.bs.1.1 2 40.19 odd 2