Properties

Label 3720.2.a.v.1.4
Level $3720$
Weight $2$
Character 3720.1
Self dual yes
Analytic conductor $29.704$
Analytic rank $0$
Dimension $5$
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] = [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 = [5,0,5,0,-5,0,3] 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: \(5\)
Coefficient field: 5.5.2294036.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 8x^{2} + 6x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.817478\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +3.33173 q^{7} +1.00000 q^{9} +6.35857 q^{11} +3.86626 q^{13} -1.00000 q^{15} +5.25814 q^{17} +2.53453 q^{19} +3.33173 q^{21} -7.99353 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.12440 q^{29} +1.00000 q^{31} +6.35857 q^{33} -3.33173 q^{35} +2.79720 q^{37} +3.86626 q^{39} +4.20085 q^{41} +4.73538 q^{43} -1.00000 q^{45} +1.63496 q^{47} +4.10043 q^{49} +5.25814 q^{51} -4.37034 q^{53} -6.35857 q^{55} +2.53453 q^{57} -0.231304 q^{59} -2.89310 q^{61} +3.33173 q^{63} -3.86626 q^{65} -13.8533 q^{67} -7.99353 q^{69} -9.42929 q^{71} -2.95492 q^{73} +1.00000 q^{75} +21.1850 q^{77} -14.4810 q^{79} +1.00000 q^{81} -14.3521 q^{83} -5.25814 q^{85} -7.12440 q^{87} +10.8195 q^{89} +12.8813 q^{91} +1.00000 q^{93} -2.53453 q^{95} +6.46261 q^{97} +6.35857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 5 q^{11} + 2 q^{13} - 5 q^{15} + 6 q^{17} + 9 q^{19} + 3 q^{21} - 3 q^{23} + 5 q^{25} + 5 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - 3 q^{35} + 4 q^{37} + 2 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 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
\(7\) 3.33173 1.25928 0.629638 − 0.776889i \(-0.283203\pi\)
0.629638 + 0.776889i \(0.283203\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.35857 1.91718 0.958591 − 0.284788i \(-0.0919231\pi\)
0.958591 + 0.284788i \(0.0919231\pi\)
\(12\) 0 0
\(13\) 3.86626 1.07231 0.536154 − 0.844120i \(-0.319877\pi\)
0.536154 + 0.844120i \(0.319877\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 5.25814 1.27529 0.637644 − 0.770331i \(-0.279909\pi\)
0.637644 + 0.770331i \(0.279909\pi\)
\(18\) 0 0
\(19\) 2.53453 0.581461 0.290730 − 0.956805i \(-0.406102\pi\)
0.290730 + 0.956805i \(0.406102\pi\)
\(20\) 0 0
\(21\) 3.33173 0.727043
\(22\) 0 0
\(23\) −7.99353 −1.66677 −0.833383 − 0.552696i \(-0.813599\pi\)
−0.833383 + 0.552696i \(0.813599\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.12440 −1.32297 −0.661484 − 0.749959i \(-0.730073\pi\)
−0.661484 + 0.749959i \(0.730073\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) 6.35857 1.10689
\(34\) 0 0
\(35\) −3.33173 −0.563165
\(36\) 0 0
\(37\) 2.79720 0.459857 0.229929 − 0.973208i \(-0.426151\pi\)
0.229929 + 0.973208i \(0.426151\pi\)
\(38\) 0 0
\(39\) 3.86626 0.619097
\(40\) 0 0
\(41\) 4.20085 0.656063 0.328031 − 0.944667i \(-0.393615\pi\)
0.328031 + 0.944667i \(0.393615\pi\)
\(42\) 0 0
\(43\) 4.73538 0.722139 0.361069 − 0.932539i \(-0.382412\pi\)
0.361069 + 0.932539i \(0.382412\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 1.63496 0.238483 0.119241 − 0.992865i \(-0.461954\pi\)
0.119241 + 0.992865i \(0.461954\pi\)
\(48\) 0 0
\(49\) 4.10043 0.585775
\(50\) 0 0
\(51\) 5.25814 0.736287
\(52\) 0 0
\(53\) −4.37034 −0.600312 −0.300156 − 0.953890i \(-0.597039\pi\)
−0.300156 + 0.953890i \(0.597039\pi\)
\(54\) 0 0
\(55\) −6.35857 −0.857389
\(56\) 0 0
\(57\) 2.53453 0.335707
\(58\) 0 0
\(59\) −0.231304 −0.0301132 −0.0150566 − 0.999887i \(-0.504793\pi\)
−0.0150566 + 0.999887i \(0.504793\pi\)
\(60\) 0 0
\(61\) −2.89310 −0.370423 −0.185212 − 0.982699i \(-0.559297\pi\)
−0.185212 + 0.982699i \(0.559297\pi\)
\(62\) 0 0
\(63\) 3.33173 0.419759
\(64\) 0 0
\(65\) −3.86626 −0.479550
\(66\) 0 0
\(67\) −13.8533 −1.69245 −0.846226 − 0.532825i \(-0.821131\pi\)
−0.846226 + 0.532825i \(0.821131\pi\)
\(68\) 0 0
\(69\) −7.99353 −0.962307
\(70\) 0 0
\(71\) −9.42929 −1.11905 −0.559526 − 0.828813i \(-0.689017\pi\)
−0.559526 + 0.828813i \(0.689017\pi\)
\(72\) 0 0
\(73\) −2.95492 −0.345847 −0.172924 − 0.984935i \(-0.555321\pi\)
−0.172924 + 0.984935i \(0.555321\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 21.1850 2.41426
\(78\) 0 0
\(79\) −14.4810 −1.62924 −0.814621 − 0.579993i \(-0.803055\pi\)
−0.814621 + 0.579993i \(0.803055\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.3521 −1.57535 −0.787674 − 0.616093i \(-0.788715\pi\)
−0.787674 + 0.616093i \(0.788715\pi\)
\(84\) 0 0
\(85\) −5.25814 −0.570326
\(86\) 0 0
\(87\) −7.12440 −0.763816
\(88\) 0 0
\(89\) 10.8195 1.14687 0.573433 − 0.819252i \(-0.305611\pi\)
0.573433 + 0.819252i \(0.305611\pi\)
\(90\) 0 0
\(91\) 12.8813 1.35033
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) −2.53453 −0.260037
\(96\) 0 0
\(97\) 6.46261 0.656178 0.328089 − 0.944647i \(-0.393595\pi\)
0.328089 + 0.944647i \(0.393595\pi\)
\(98\) 0 0
\(99\) 6.35857 0.639060
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.v.1.4 ✓ 5
4.3 odd 2 7440.2.a.cd.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.v.1.4 ✓ 5 1.1 even 1 trivial
7440.2.a.cd.1.2 5 4.3 odd 2