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Properties

Label 7360.2.a.ce.1.1
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
Analytic rank $0$
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 = [3,0,1,0,3,0,-3,0,10,0,3] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) 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 1.1
Root \(-3.11903\) 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)
 
\(f(q)\) \(=\) \(q-3.11903 q^{3} +1.00000 q^{5} -4.50973 q^{7} +6.72833 q^{9} +4.33763 q^{11} +3.72833 q^{13} -3.11903 q^{15} +1.11903 q^{17} +4.50973 q^{19} +14.0660 q^{21} +1.00000 q^{23} +1.00000 q^{25} -11.6288 q^{27} +8.23805 q^{29} -1.72833 q^{31} -13.5292 q^{33} -4.50973 q^{35} +0.781399 q^{37} -11.6288 q^{39} +3.90043 q^{41} +8.00000 q^{43} +6.72833 q^{45} +11.4567 q^{47} +13.3376 q^{49} -3.49027 q^{51} +6.00000 q^{53} +4.33763 q^{55} -14.0660 q^{57} -2.23805 q^{59} -3.55623 q^{61} -30.3429 q^{63} +3.72833 q^{65} +2.43720 q^{67} -3.11903 q^{69} -7.11903 q^{71} -9.45665 q^{73} -3.11903 q^{75} -19.5615 q^{77} +14.9133 q^{79} +16.0854 q^{81} +2.78140 q^{83} +1.11903 q^{85} -25.6947 q^{87} -7.69471 q^{89} -16.8137 q^{91} +5.39070 q^{93} +4.50973 q^{95} -0.642920 q^{97} +29.1850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 10 q^{9} + 3 q^{11} + q^{13} + q^{15} - 7 q^{17} + 3 q^{19} + 22 q^{21} + 3 q^{23} + 3 q^{25} - 14 q^{27} + 4 q^{29} + 5 q^{31} - 9 q^{33} - 3 q^{35} + 2 q^{37} - 14 q^{39}+ \cdots + 57 q^{99}+O(q^{100}) \) 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
\(3\) −3.11903 −1.80077 −0.900385 − 0.435093i \(-0.856715\pi\)
−0.900385 + 0.435093i \(0.856715\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.50973 −1.70452 −0.852258 − 0.523122i \(-0.824767\pi\)
−0.852258 + 0.523122i \(0.824767\pi\)
\(8\) 0 0
\(9\) 6.72833 2.24278
\(10\) 0 0
\(11\) 4.33763 1.30784 0.653922 − 0.756562i \(-0.273122\pi\)
0.653922 + 0.756562i \(0.273122\pi\)
\(12\) 0 0
\(13\) 3.72833 1.03405 0.517026 − 0.855970i \(-0.327039\pi\)
0.517026 + 0.855970i \(0.327039\pi\)
\(14\) 0 0
\(15\) −3.11903 −0.805329
\(16\) 0 0
\(17\) 1.11903 0.271404 0.135702 − 0.990750i \(-0.456671\pi\)
0.135702 + 0.990750i \(0.456671\pi\)
\(18\) 0 0
\(19\) 4.50973 1.03460 0.517301 − 0.855803i \(-0.326937\pi\)
0.517301 + 0.855803i \(0.326937\pi\)
\(20\) 0 0
\(21\) 14.0660 3.06944
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −11.6288 −2.23795
\(28\) 0 0
\(29\) 8.23805 1.52977 0.764884 − 0.644168i \(-0.222796\pi\)
0.764884 + 0.644168i \(0.222796\pi\)
\(30\) 0 0
\(31\) −1.72833 −0.310417 −0.155208 − 0.987882i \(-0.549605\pi\)
−0.155208 + 0.987882i \(0.549605\pi\)
\(32\) 0 0
\(33\) −13.5292 −2.35513
\(34\) 0 0
\(35\) −4.50973 −0.762283
\(36\) 0 0
\(37\) 0.781399 0.128461 0.0642306 − 0.997935i \(-0.479541\pi\)
0.0642306 + 0.997935i \(0.479541\pi\)
\(38\) 0 0
\(39\) −11.6288 −1.86209
\(40\) 0 0
\(41\) 3.90043 0.609144 0.304572 − 0.952489i \(-0.401487\pi\)
0.304572 + 0.952489i \(0.401487\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 6.72833 1.00300
\(46\) 0 0
\(47\) 11.4567 1.67112 0.835562 − 0.549396i \(-0.185142\pi\)
0.835562 + 0.549396i \(0.185142\pi\)
\(48\) 0 0
\(49\) 13.3376 1.90538
\(50\) 0 0
\(51\) −3.49027 −0.488736
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 4.33763 0.584886
\(56\) 0 0
\(57\) −14.0660 −1.86308
\(58\) 0 0
\(59\) −2.23805 −0.291370 −0.145685 − 0.989331i \(-0.546539\pi\)
−0.145685 + 0.989331i \(0.546539\pi\)
\(60\) 0 0
\(61\) −3.55623 −0.455329 −0.227664 − 0.973740i \(-0.573109\pi\)
−0.227664 + 0.973740i \(0.573109\pi\)
\(62\) 0 0
\(63\) −30.3429 −3.82285
\(64\) 0 0
\(65\) 3.72833 0.462442
\(66\) 0 0
\(67\) 2.43720 0.297752 0.148876 − 0.988856i \(-0.452435\pi\)
0.148876 + 0.988856i \(0.452435\pi\)
\(68\) 0 0
\(69\) −3.11903 −0.375487
\(70\) 0 0
\(71\) −7.11903 −0.844873 −0.422437 − 0.906393i \(-0.638825\pi\)
−0.422437 + 0.906393i \(0.638825\pi\)
\(72\) 0 0
\(73\) −9.45665 −1.10682 −0.553409 − 0.832910i \(-0.686673\pi\)
−0.553409 + 0.832910i \(0.686673\pi\)
\(74\) 0 0
\(75\) −3.11903 −0.360154
\(76\) 0 0
\(77\) −19.5615 −2.22924
\(78\) 0 0
\(79\) 14.9133 1.67788 0.838939 − 0.544225i \(-0.183176\pi\)
0.838939 + 0.544225i \(0.183176\pi\)
\(80\) 0 0
\(81\) 16.0854 1.78727
\(82\) 0 0
\(83\) 2.78140 0.305298 0.152649 − 0.988280i \(-0.451220\pi\)
0.152649 + 0.988280i \(0.451220\pi\)
\(84\) 0 0
\(85\) 1.11903 0.121375
\(86\) 0 0
\(87\) −25.6947 −2.75476
\(88\) 0 0
\(89\) −7.69471 −0.815637 −0.407819 − 0.913063i \(-0.633710\pi\)
−0.407819 + 0.913063i \(0.633710\pi\)
\(90\) 0 0
\(91\) −16.8137 −1.76256
\(92\) 0 0
\(93\) 5.39070 0.558989
\(94\) 0 0
\(95\) 4.50973 0.462688
\(96\) 0 0
\(97\) −0.642920 −0.0652786 −0.0326393 − 0.999467i \(-0.510391\pi\)
−0.0326393 + 0.999467i \(0.510391\pi\)
\(98\) 0 0
\(99\) 29.1850 2.93320
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.ce.1.1 3
4.3 odd 2 7360.2.a.bz.1.3 3
8.3 odd 2 230.2.a.d.1.1 ✓ 3
8.5 even 2 1840.2.a.r.1.3 3
24.11 even 2 2070.2.a.z.1.3 3
40.3 even 4 1150.2.b.j.599.1 6
40.19 odd 2 1150.2.a.q.1.3 3
40.27 even 4 1150.2.b.j.599.6 6
40.29 even 2 9200.2.a.cf.1.1 3
184.91 even 2 5290.2.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 ✓ 3 8.3 odd 2
1150.2.a.q.1.3 3 40.19 odd 2
1150.2.b.j.599.1 6 40.3 even 4
1150.2.b.j.599.6 6 40.27 even 4
1840.2.a.r.1.3 3 8.5 even 2
2070.2.a.z.1.3 3 24.11 even 2
5290.2.a.r.1.1 3 184.91 even 2
7360.2.a.bz.1.3 3 4.3 odd 2
7360.2.a.ce.1.1 3 1.1 even 1 trivial
9200.2.a.cf.1.1 3 40.29 even 2