Properties

Label 3720.2.a.l.1.2
Level $3720$
Weight $2$
Character 3720.1
Self dual yes
Analytic conductor $29.704$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3720,2,Mod(1,3720)] 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("3720.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3720.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,-3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.7043495519\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.65544\) of defining polynomial
Character \(\chi\) \(=\) 3720.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +0.259511 q^{7} +1.00000 q^{9} +4.00000 q^{11} +2.25951 q^{13} +1.00000 q^{15} +3.31088 q^{17} +6.10275 q^{19} -0.259511 q^{21} -0.791864 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.53235 q^{29} -1.00000 q^{31} -4.00000 q^{33} -0.259511 q^{35} -4.36226 q^{37} -2.25951 q^{39} +10.2055 q^{41} -10.1027 q^{43} -1.00000 q^{45} -7.41363 q^{47} -6.93265 q^{49} -3.31088 q^{51} -0.689115 q^{53} -4.00000 q^{55} -6.10275 q^{57} +5.05137 q^{59} -1.48098 q^{61} +0.259511 q^{63} -2.25951 q^{65} +5.84324 q^{67} +0.791864 q^{69} +9.05137 q^{71} +0.362259 q^{73} -1.00000 q^{75} +1.03804 q^{77} +4.79186 q^{79} +1.00000 q^{81} +18.0354 q^{83} -3.31088 q^{85} +2.53235 q^{87} -9.15412 q^{89} +0.586367 q^{91} +1.00000 q^{93} -6.10275 q^{95} +9.68648 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} + 12 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} + 2 q^{21} + 4 q^{23} + 3 q^{25} - 3 q^{27} - 4 q^{29} - 3 q^{31} - 12 q^{33} + 2 q^{35} + 8 q^{37} - 4 q^{39} - 6 q^{41}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.259511 0.0980858 0.0490429 − 0.998797i \(-0.484383\pi\)
0.0490429 + 0.998797i \(0.484383\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 2.25951 0.626675 0.313338 − 0.949642i \(-0.398553\pi\)
0.313338 + 0.949642i \(0.398553\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 3.31088 0.803008 0.401504 − 0.915857i \(-0.368488\pi\)
0.401504 + 0.915857i \(0.368488\pi\)
\(18\) 0 0
\(19\) 6.10275 1.40007 0.700033 − 0.714110i \(-0.253168\pi\)
0.700033 + 0.714110i \(0.253168\pi\)
\(20\) 0 0
\(21\) −0.259511 −0.0566298
\(22\) 0 0
\(23\) −0.791864 −0.165115 −0.0825575 − 0.996586i \(-0.526309\pi\)
−0.0825575 + 0.996586i \(0.526309\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.53235 −0.470246 −0.235123 − 0.971966i \(-0.575549\pi\)
−0.235123 + 0.971966i \(0.575549\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −0.259511 −0.0438653
\(36\) 0 0
\(37\) −4.36226 −0.717151 −0.358575 − 0.933501i \(-0.616737\pi\)
−0.358575 + 0.933501i \(0.616737\pi\)
\(38\) 0 0
\(39\) −2.25951 −0.361811
\(40\) 0 0
\(41\) 10.2055 1.59383 0.796915 − 0.604091i \(-0.206464\pi\)
0.796915 + 0.604091i \(0.206464\pi\)
\(42\) 0 0
\(43\) −10.1027 −1.54065 −0.770327 − 0.637649i \(-0.779907\pi\)
−0.770327 + 0.637649i \(0.779907\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −7.41363 −1.08139 −0.540695 − 0.841219i \(-0.681839\pi\)
−0.540695 + 0.841219i \(0.681839\pi\)
\(48\) 0 0
\(49\) −6.93265 −0.990379
\(50\) 0 0
\(51\) −3.31088 −0.463617
\(52\) 0 0
\(53\) −0.689115 −0.0946573 −0.0473286 − 0.998879i \(-0.515071\pi\)
−0.0473286 + 0.998879i \(0.515071\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −6.10275 −0.808329
\(58\) 0 0
\(59\) 5.05137 0.657633 0.328816 − 0.944394i \(-0.393350\pi\)
0.328816 + 0.944394i \(0.393350\pi\)
\(60\) 0 0
\(61\) −1.48098 −0.189620 −0.0948100 − 0.995495i \(-0.530224\pi\)
−0.0948100 + 0.995495i \(0.530224\pi\)
\(62\) 0 0
\(63\) 0.259511 0.0326953
\(64\) 0 0
\(65\) −2.25951 −0.280258
\(66\) 0 0
\(67\) 5.84324 0.713865 0.356933 − 0.934130i \(-0.383823\pi\)
0.356933 + 0.934130i \(0.383823\pi\)
\(68\) 0 0
\(69\) 0.791864 0.0953292
\(70\) 0 0
\(71\) 9.05137 1.07420 0.537100 − 0.843518i \(-0.319520\pi\)
0.537100 + 0.843518i \(0.319520\pi\)
\(72\) 0 0
\(73\) 0.362259 0.0423992 0.0211996 − 0.999775i \(-0.493251\pi\)
0.0211996 + 0.999775i \(0.493251\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 1.03804 0.118296
\(78\) 0 0
\(79\) 4.79186 0.539127 0.269563 − 0.962983i \(-0.413121\pi\)
0.269563 + 0.962983i \(0.413121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 18.0354 1.97964 0.989821 − 0.142316i \(-0.0454548\pi\)
0.989821 + 0.142316i \(0.0454548\pi\)
\(84\) 0 0
\(85\) −3.31088 −0.359116
\(86\) 0 0
\(87\) 2.53235 0.271497
\(88\) 0 0
\(89\) −9.15412 −0.970335 −0.485168 − 0.874421i \(-0.661241\pi\)
−0.485168 + 0.874421i \(0.661241\pi\)
\(90\) 0 0
\(91\) 0.586367 0.0614679
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) −6.10275 −0.626129
\(96\) 0 0
\(97\) 9.68648 0.983513 0.491756 − 0.870733i \(-0.336355\pi\)
0.491756 + 0.870733i \(0.336355\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 3720.2.a.l.1.2 ✓ 3
4.3 odd 2 7440.2.a.bu.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.l.1.2 ✓ 3 1.1 even 1 trivial
7440.2.a.bu.1.2 3 4.3 odd 2