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Properties

Label 1360.4.a.a.1.1
Level $1360$
Weight $4$
Character 1360.1
Self dual yes
Analytic conductor $80.243$
Analytic rank $1$
Dimension $1$
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] = [1360,4,Mod(1,1360)] 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("1360.1"); S:= CuspForms(chi, 4); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1360.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-10,0,5,0,22,0,73,0,30] 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: \(80.2425976078\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1360.1

$q$-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)
 
\(f(q)\) \(=\) \(q-10.0000 q^{3} +5.00000 q^{5} +22.0000 q^{7} +73.0000 q^{9} +30.0000 q^{11} -46.0000 q^{13} -50.0000 q^{15} +17.0000 q^{17} -104.000 q^{19} -220.000 q^{21} -42.0000 q^{23} +25.0000 q^{25} -460.000 q^{27} -66.0000 q^{29} -194.000 q^{31} -300.000 q^{33} +110.000 q^{35} +206.000 q^{37} +460.000 q^{39} -126.000 q^{41} +388.000 q^{43} +365.000 q^{45} +540.000 q^{47} +141.000 q^{49} -170.000 q^{51} +78.0000 q^{53} +150.000 q^{55} +1040.00 q^{57} -432.000 q^{59} -610.000 q^{61} +1606.00 q^{63} -230.000 q^{65} -848.000 q^{67} +420.000 q^{69} +174.000 q^{71} +362.000 q^{73} -250.000 q^{75} +660.000 q^{77} -398.000 q^{79} +2629.00 q^{81} -828.000 q^{83} +85.0000 q^{85} +660.000 q^{87} +630.000 q^{89} -1012.00 q^{91} +1940.00 q^{93} -520.000 q^{95} -1486.00 q^{97} +2190.00 q^{99} +O(q^{100})\)

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
\(3\) −10.0000 −1.92450 −0.962250 − 0.272166i \(-0.912260\pi\)
−0.962250 + 0.272166i \(0.912260\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 22.0000 1.18789 0.593944 − 0.804506i \(-0.297570\pi\)
0.593944 + 0.804506i \(0.297570\pi\)
\(8\) 0 0
\(9\) 73.0000 2.70370
\(10\) 0 0
\(11\) 30.0000 0.822304 0.411152 − 0.911567i \(-0.365127\pi\)
0.411152 + 0.911567i \(0.365127\pi\)
\(12\) 0 0
\(13\) −46.0000 −0.981393 −0.490696 − 0.871331i \(-0.663258\pi\)
−0.490696 + 0.871331i \(0.663258\pi\)
\(14\) 0 0
\(15\) −50.0000 −0.860663
\(16\) 0 0
\(17\) 17.0000 0.242536
\(18\) 0 0
\(19\) −104.000 −1.25575 −0.627875 − 0.778314i \(-0.716075\pi\)
−0.627875 + 0.778314i \(0.716075\pi\)
\(20\) 0 0
\(21\) −220.000 −2.28609
\(22\) 0 0
\(23\) −42.0000 −0.380765 −0.190383 − 0.981710i \(-0.560973\pi\)
−0.190383 + 0.981710i \(0.560973\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −460.000 −3.27878
\(28\) 0 0
\(29\) −66.0000 −0.422617 −0.211308 − 0.977419i \(-0.567772\pi\)
−0.211308 + 0.977419i \(0.567772\pi\)
\(30\) 0 0
\(31\) −194.000 −1.12398 −0.561991 − 0.827143i \(-0.689964\pi\)
−0.561991 + 0.827143i \(0.689964\pi\)
\(32\) 0 0
\(33\) −300.000 −1.58252
\(34\) 0 0
\(35\) 110.000 0.531240
\(36\) 0 0
\(37\) 206.000 0.915302 0.457651 − 0.889132i \(-0.348691\pi\)
0.457651 + 0.889132i \(0.348691\pi\)
\(38\) 0 0
\(39\) 460.000 1.88869
\(40\) 0 0
\(41\) −126.000 −0.479949 −0.239974 − 0.970779i \(-0.577139\pi\)
−0.239974 + 0.970779i \(0.577139\pi\)
\(42\) 0 0
\(43\) 388.000 1.37603 0.688017 − 0.725695i \(-0.258482\pi\)
0.688017 + 0.725695i \(0.258482\pi\)
\(44\) 0 0
\(45\) 365.000 1.20913
\(46\) 0 0
\(47\) 540.000 1.67590 0.837948 − 0.545750i \(-0.183755\pi\)
0.837948 + 0.545750i \(0.183755\pi\)
\(48\) 0 0
\(49\) 141.000 0.411079
\(50\) 0 0
\(51\) −170.000 −0.466760
\(52\) 0 0
\(53\) 78.0000 0.202153 0.101077 − 0.994879i \(-0.467771\pi\)
0.101077 + 0.994879i \(0.467771\pi\)
\(54\) 0 0
\(55\) 150.000 0.367745
\(56\) 0 0
\(57\) 1040.00 2.41669
\(58\) 0 0
\(59\) −432.000 −0.953248 −0.476624 − 0.879107i \(-0.658140\pi\)
−0.476624 + 0.879107i \(0.658140\pi\)
\(60\) 0 0
\(61\) −610.000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 1606.00 3.21170
\(64\) 0 0
\(65\) −230.000 −0.438892
\(66\) 0 0
\(67\) −848.000 −1.54626 −0.773132 − 0.634245i \(-0.781311\pi\)
−0.773132 + 0.634245i \(0.781311\pi\)
\(68\) 0 0
\(69\) 420.000 0.732783
\(70\) 0 0
\(71\) 174.000 0.290845 0.145423 − 0.989370i \(-0.453546\pi\)
0.145423 + 0.989370i \(0.453546\pi\)
\(72\) 0 0
\(73\) 362.000 0.580396 0.290198 − 0.956967i \(-0.406279\pi\)
0.290198 + 0.956967i \(0.406279\pi\)
\(74\) 0 0
\(75\) −250.000 −0.384900
\(76\) 0 0
\(77\) 660.000 0.976805
\(78\) 0 0
\(79\) −398.000 −0.566816 −0.283408 − 0.958999i \(-0.591465\pi\)
−0.283408 + 0.958999i \(0.591465\pi\)
\(80\) 0 0
\(81\) 2629.00 3.60631
\(82\) 0 0
\(83\) −828.000 −1.09500 −0.547499 − 0.836806i \(-0.684420\pi\)
−0.547499 + 0.836806i \(0.684420\pi\)
\(84\) 0 0
\(85\) 85.0000 0.108465
\(86\) 0 0
\(87\) 660.000 0.813327
\(88\) 0 0
\(89\) 630.000 0.750336 0.375168 − 0.926957i \(-0.377585\pi\)
0.375168 + 0.926957i \(0.377585\pi\)
\(90\) 0 0
\(91\) −1012.00 −1.16578
\(92\) 0 0
\(93\) 1940.00 2.16310
\(94\) 0 0
\(95\) −520.000 −0.561588
\(96\) 0 0
\(97\) −1486.00 −1.55547 −0.777734 − 0.628593i \(-0.783631\pi\)
−0.777734 + 0.628593i \(0.783631\pi\)
\(98\) 0 0
\(99\) 2190.00 2.22327
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 1360.4.a.a.1.1 1
4.3 odd 2 85.4.a.c.1.1 ✓ 1
12.11 even 2 765.4.a.a.1.1 1
20.3 even 4 425.4.b.d.324.1 2
20.7 even 4 425.4.b.d.324.2 2
20.19 odd 2 425.4.a.a.1.1 1
68.67 odd 2 1445.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.c.1.1 ✓ 1 4.3 odd 2
425.4.a.a.1.1 1 20.19 odd 2
425.4.b.d.324.1 2 20.3 even 4
425.4.b.d.324.2 2 20.7 even 4
765.4.a.a.1.1 1 12.11 even 2
1360.4.a.a.1.1 1 1.1 even 1 trivial
1445.4.a.f.1.1 1 68.67 odd 2