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Properties

Label 1360.2.c.f.1121.2
Level $1360$
Weight $2$
Character 1360.1121
Analytic conductor $10.860$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [1360,2,Mod(1121,1360)] 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("1360.1121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1360.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,2,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.8596546749\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1121.2
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 1360.1121
Dual form 1360.2.c.f.1121.5

$q$-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)
 
\(f(q)\) \(=\) \(q-1.67513i q^{3} -1.00000i q^{5} +1.28726i q^{7} +0.193937 q^{9} +0.481194i q^{11} -2.15633 q^{13} -1.67513 q^{15} +(-1.86907 - 3.67513i) q^{17} -3.35026 q^{19} +2.15633 q^{21} -8.24965i q^{23} -1.00000 q^{25} -5.35026i q^{27} +0.649738i q^{29} -1.83146i q^{31} +0.806063 q^{33} +1.28726 q^{35} +4.31265i q^{37} +3.61213i q^{39} -11.2750i q^{41} -8.15633 q^{43} -0.193937i q^{45} +6.54420 q^{47} +5.34297 q^{49} +(-6.15633 + 3.13093i) q^{51} -8.57452 q^{53} +0.481194 q^{55} +5.61213i q^{57} -4.96239 q^{59} +2.83638i q^{61} +0.249646i q^{63} +2.15633i q^{65} -4.93207 q^{67} -13.8192 q^{69} +14.5320i q^{71} -13.3503i q^{73} +1.67513i q^{75} -0.619421 q^{77} -9.05571i q^{79} -8.38058 q^{81} +13.4314 q^{83} +(-3.67513 + 1.86907i) q^{85} +1.08840 q^{87} -16.7816 q^{89} -2.77575i q^{91} -3.06793 q^{93} +3.35026i q^{95} -3.66291i q^{97} +0.0933212i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{9} + 8 q^{13} - 2 q^{17} - 8 q^{21} - 6 q^{25} + 4 q^{33} - 4 q^{35} - 28 q^{43} + 20 q^{47} - 14 q^{49} - 16 q^{51} - 28 q^{53} - 8 q^{55} - 8 q^{59} - 12 q^{67} - 28 q^{77} - 26 q^{81} - 4 q^{83}+ \cdots - 36 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(341\) \(511\) \(817\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) − 1.67513i − 0.967137i −0.875306 − 0.483569i \(-0.839340\pi\)
0.875306 − 0.483569i \(-0.160660\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 1.28726i 0.486538i 0.969959 + 0.243269i \(0.0782197\pi\)
−0.969959 + 0.243269i \(0.921780\pi\)
\(8\) 0 0
\(9\) 0.193937 0.0646455
\(10\) 0 0
\(11\) 0.481194i 0.145086i 0.997365 + 0.0725428i \(0.0231114\pi\)
−0.997365 + 0.0725428i \(0.976889\pi\)
\(12\) 0 0
\(13\) −2.15633 −0.598057 −0.299028 − 0.954244i \(-0.596663\pi\)
−0.299028 + 0.954244i \(0.596663\pi\)
\(14\) 0 0
\(15\) −1.67513 −0.432517
\(16\) 0 0
\(17\) −1.86907 − 3.67513i −0.453315 − 0.891350i
\(18\) 0 0
\(19\) −3.35026 −0.768603 −0.384301 − 0.923208i \(-0.625558\pi\)
−0.384301 + 0.923208i \(0.625558\pi\)
\(20\) 0 0
\(21\) 2.15633 0.470549
\(22\) 0 0
\(23\) − 8.24965i − 1.72017i −0.510151 − 0.860085i \(-0.670410\pi\)
0.510151 − 0.860085i \(-0.329590\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) − 5.35026i − 1.02966i
\(28\) 0 0
\(29\) 0.649738i 0.120653i 0.998179 + 0.0603267i \(0.0192142\pi\)
−0.998179 + 0.0603267i \(0.980786\pi\)
\(30\) 0 0
\(31\) − 1.83146i − 0.328939i −0.986382 − 0.164470i \(-0.947409\pi\)
0.986382 − 0.164470i \(-0.0525912\pi\)
\(32\) 0 0
\(33\) 0.806063 0.140318
\(34\) 0 0
\(35\) 1.28726 0.217586
\(36\) 0 0
\(37\) 4.31265i 0.708995i 0.935057 + 0.354498i \(0.115348\pi\)
−0.935057 + 0.354498i \(0.884652\pi\)
\(38\) 0 0
\(39\) 3.61213i 0.578403i
\(40\) 0 0
\(41\) − 11.2750i − 1.76087i −0.474171 − 0.880433i \(-0.657252\pi\)
0.474171 − 0.880433i \(-0.342748\pi\)
\(42\) 0 0
\(43\) −8.15633 −1.24383 −0.621914 − 0.783086i \(-0.713645\pi\)
−0.621914 + 0.783086i \(0.713645\pi\)
\(44\) 0 0
\(45\) − 0.193937i − 0.0289104i
\(46\) 0 0
\(47\) 6.54420 0.954569 0.477285 − 0.878749i \(-0.341621\pi\)
0.477285 + 0.878749i \(0.341621\pi\)
\(48\) 0 0
\(49\) 5.34297 0.763281
\(50\) 0 0
\(51\) −6.15633 + 3.13093i −0.862058 + 0.438418i
\(52\) 0 0
\(53\) −8.57452 −1.17780 −0.588900 − 0.808206i \(-0.700439\pi\)
−0.588900 + 0.808206i \(0.700439\pi\)
\(54\) 0 0
\(55\) 0.481194 0.0648842
\(56\) 0 0
\(57\) 5.61213i 0.743344i
\(58\) 0 0
\(59\) −4.96239 −0.646048 −0.323024 − 0.946391i \(-0.604699\pi\)
−0.323024 + 0.946391i \(0.604699\pi\)
\(60\) 0 0
\(61\) 2.83638i 0.363161i 0.983376 + 0.181581i \(0.0581213\pi\)
−0.983376 + 0.181581i \(0.941879\pi\)
\(62\) 0 0
\(63\) 0.249646i 0.0314525i
\(64\) 0 0
\(65\) 2.15633i 0.267459i
\(66\) 0 0
\(67\) −4.93207 −0.602548 −0.301274 − 0.953538i \(-0.597412\pi\)
−0.301274 + 0.953538i \(0.597412\pi\)
\(68\) 0 0
\(69\) −13.8192 −1.66364
\(70\) 0 0
\(71\) 14.5320i 1.72463i 0.506373 + 0.862314i \(0.330986\pi\)
−0.506373 + 0.862314i \(0.669014\pi\)
\(72\) 0 0
\(73\) − 13.3503i − 1.56253i −0.624200 − 0.781265i \(-0.714575\pi\)
0.624200 − 0.781265i \(-0.285425\pi\)
\(74\) 0 0
\(75\) 1.67513i 0.193427i
\(76\) 0 0
\(77\) −0.619421 −0.0705896
\(78\) 0 0
\(79\) − 9.05571i − 1.01885i −0.860516 − 0.509423i \(-0.829859\pi\)
0.860516 − 0.509423i \(-0.170141\pi\)
\(80\) 0 0
\(81\) −8.38058 −0.931175
\(82\) 0 0
\(83\) 13.4314 1.47428 0.737142 − 0.675738i \(-0.236175\pi\)
0.737142 + 0.675738i \(0.236175\pi\)
\(84\) 0 0
\(85\) −3.67513 + 1.86907i −0.398624 + 0.202729i
\(86\) 0 0
\(87\) 1.08840 0.116688
\(88\) 0 0
\(89\) −16.7816 −1.77885 −0.889424 − 0.457082i \(-0.848894\pi\)
−0.889424 + 0.457082i \(0.848894\pi\)
\(90\) 0 0
\(91\) − 2.77575i − 0.290977i
\(92\) 0 0
\(93\) −3.06793 −0.318129
\(94\) 0 0
\(95\) 3.35026i 0.343730i
\(96\) 0 0
\(97\) − 3.66291i − 0.371912i −0.982558 − 0.185956i \(-0.940462\pi\)
0.982558 − 0.185956i \(-0.0595383\pi\)
\(98\) 0 0
\(99\) 0.0933212i 0.00937913i
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 1360.2.c.f.1121.2 6
4.3 odd 2 85.2.d.a.16.6 yes 6
12.11 even 2 765.2.g.b.271.2 6
17.16 even 2 inner 1360.2.c.f.1121.5 6
20.3 even 4 425.2.c.b.424.2 6
20.7 even 4 425.2.c.a.424.5 6
20.19 odd 2 425.2.d.c.101.1 6
68.47 odd 4 1445.2.a.k.1.1 3
68.55 odd 4 1445.2.a.j.1.1 3
68.67 odd 2 85.2.d.a.16.5 ✓ 6
204.203 even 2 765.2.g.b.271.1 6
340.67 even 4 425.2.c.b.424.5 6
340.203 even 4 425.2.c.a.424.2 6
340.259 odd 4 7225.2.a.q.1.3 3
340.319 odd 4 7225.2.a.r.1.3 3
340.339 odd 2 425.2.d.c.101.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.d.a.16.5 ✓ 6 68.67 odd 2
85.2.d.a.16.6 yes 6 4.3 odd 2
425.2.c.a.424.2 6 340.203 even 4
425.2.c.a.424.5 6 20.7 even 4
425.2.c.b.424.2 6 20.3 even 4
425.2.c.b.424.5 6 340.67 even 4
425.2.d.c.101.1 6 20.19 odd 2
425.2.d.c.101.2 6 340.339 odd 2
765.2.g.b.271.1 6 204.203 even 2
765.2.g.b.271.2 6 12.11 even 2
1360.2.c.f.1121.2 6 1.1 even 1 trivial
1360.2.c.f.1121.5 6 17.16 even 2 inner
1445.2.a.j.1.1 3 68.55 odd 4
1445.2.a.k.1.1 3 68.47 odd 4
7225.2.a.q.1.3 3 340.259 odd 4
7225.2.a.r.1.3 3 340.319 odd 4