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Properties

Label 2640.2.a.be.1.3
Level $2640$
Weight $2$
Character 2640.1
Self dual yes
Analytic conductor $21.081$
Analytic rank $1$
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] = [2640,2,Mod(1,2640)] 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("2640.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2640.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,3,0,0,0,3,0,-3,0,-2,0,-3,0,-2,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(21.0805061336\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 2640.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +4.42864 q^{7} +1.00000 q^{9} -1.00000 q^{11} -0.622216 q^{13} -1.00000 q^{15} -5.18421 q^{17} -7.05086 q^{19} -4.42864 q^{21} -8.85728 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.80642 q^{29} -2.75557 q^{31} +1.00000 q^{33} +4.42864 q^{35} -2.00000 q^{37} +0.622216 q^{39} -0.193576 q^{41} -5.67307 q^{43} +1.00000 q^{45} +2.75557 q^{47} +12.6128 q^{49} +5.18421 q^{51} -10.8573 q^{53} -1.00000 q^{55} +7.05086 q^{57} +4.85728 q^{59} +6.85728 q^{61} +4.42864 q^{63} -0.622216 q^{65} +1.24443 q^{67} +8.85728 q^{69} -2.75557 q^{71} +4.23506 q^{73} -1.00000 q^{75} -4.42864 q^{77} -8.56199 q^{79} +1.00000 q^{81} -0.133353 q^{83} -5.18421 q^{85} +7.80642 q^{87} +5.61285 q^{89} -2.75557 q^{91} +2.75557 q^{93} -7.05086 q^{95} +7.24443 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{15} - 2 q^{17} - 8 q^{19} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{37} + 2 q^{39} - 14 q^{41} - 4 q^{43} + 3 q^{45} + 8 q^{47}+ \cdots - 3 q^{99}+O(q^{100}) \) 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
\(7\) 4.42864 1.67387 0.836934 − 0.547304i \(-0.184346\pi\)
0.836934 + 0.547304i \(0.184346\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −0.622216 −0.172572 −0.0862858 − 0.996270i \(-0.527500\pi\)
−0.0862858 + 0.996270i \(0.527500\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −5.18421 −1.25736 −0.628678 − 0.777666i \(-0.716403\pi\)
−0.628678 + 0.777666i \(0.716403\pi\)
\(18\) 0 0
\(19\) −7.05086 −1.61758 −0.808789 − 0.588100i \(-0.799876\pi\)
−0.808789 + 0.588100i \(0.799876\pi\)
\(20\) 0 0
\(21\) −4.42864 −0.966408
\(22\) 0 0
\(23\) −8.85728 −1.84687 −0.923435 − 0.383754i \(-0.874631\pi\)
−0.923435 + 0.383754i \(0.874631\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.80642 −1.44962 −0.724808 − 0.688951i \(-0.758072\pi\)
−0.724808 + 0.688951i \(0.758072\pi\)
\(30\) 0 0
\(31\) −2.75557 −0.494915 −0.247457 − 0.968899i \(-0.579595\pi\)
−0.247457 + 0.968899i \(0.579595\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 4.42864 0.748577
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0.622216 0.0996342
\(40\) 0 0
\(41\) −0.193576 −0.0302315 −0.0151158 − 0.999886i \(-0.504812\pi\)
−0.0151158 + 0.999886i \(0.504812\pi\)
\(42\) 0 0
\(43\) −5.67307 −0.865135 −0.432568 − 0.901602i \(-0.642392\pi\)
−0.432568 + 0.901602i \(0.642392\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 2.75557 0.401941 0.200971 − 0.979597i \(-0.435590\pi\)
0.200971 + 0.979597i \(0.435590\pi\)
\(48\) 0 0
\(49\) 12.6128 1.80184
\(50\) 0 0
\(51\) 5.18421 0.725934
\(52\) 0 0
\(53\) −10.8573 −1.49136 −0.745681 − 0.666303i \(-0.767876\pi\)
−0.745681 + 0.666303i \(0.767876\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 7.05086 0.933909
\(58\) 0 0
\(59\) 4.85728 0.632364 0.316182 − 0.948699i \(-0.397599\pi\)
0.316182 + 0.948699i \(0.397599\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 − 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) 0 0
\(63\) 4.42864 0.557956
\(64\) 0 0
\(65\) −0.622216 −0.0771764
\(66\) 0 0
\(67\) 1.24443 0.152031 0.0760157 − 0.997107i \(-0.475780\pi\)
0.0760157 + 0.997107i \(0.475780\pi\)
\(68\) 0 0
\(69\) 8.85728 1.06629
\(70\) 0 0
\(71\) −2.75557 −0.327026 −0.163513 − 0.986541i \(-0.552283\pi\)
−0.163513 + 0.986541i \(0.552283\pi\)
\(72\) 0 0
\(73\) 4.23506 0.495677 0.247838 − 0.968801i \(-0.420280\pi\)
0.247838 + 0.968801i \(0.420280\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −4.42864 −0.504690
\(78\) 0 0
\(79\) −8.56199 −0.963299 −0.481650 − 0.876364i \(-0.659962\pi\)
−0.481650 + 0.876364i \(0.659962\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.133353 −0.0146374 −0.00731870 − 0.999973i \(-0.502330\pi\)
−0.00731870 + 0.999973i \(0.502330\pi\)
\(84\) 0 0
\(85\) −5.18421 −0.562306
\(86\) 0 0
\(87\) 7.80642 0.836936
\(88\) 0 0
\(89\) 5.61285 0.594961 0.297480 − 0.954728i \(-0.403854\pi\)
0.297480 + 0.954728i \(0.403854\pi\)
\(90\) 0 0
\(91\) −2.75557 −0.288862
\(92\) 0 0
\(93\) 2.75557 0.285739
\(94\) 0 0
\(95\) −7.05086 −0.723402
\(96\) 0 0
\(97\) 7.24443 0.735561 0.367780 − 0.929913i \(-0.380118\pi\)
0.367780 + 0.929913i \(0.380118\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
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 2640.2.a.be.1.3 3
3.2 odd 2 7920.2.a.cj.1.3 3
4.3 odd 2 165.2.a.c.1.3 ✓ 3
12.11 even 2 495.2.a.e.1.1 3
20.3 even 4 825.2.c.g.199.2 6
20.7 even 4 825.2.c.g.199.5 6
20.19 odd 2 825.2.a.k.1.1 3
28.27 even 2 8085.2.a.bk.1.3 3
44.43 even 2 1815.2.a.m.1.1 3
60.23 odd 4 2475.2.c.r.199.5 6
60.47 odd 4 2475.2.c.r.199.2 6
60.59 even 2 2475.2.a.bb.1.3 3
132.131 odd 2 5445.2.a.z.1.3 3
220.219 even 2 9075.2.a.cf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 ✓ 3 4.3 odd 2
495.2.a.e.1.1 3 12.11 even 2
825.2.a.k.1.1 3 20.19 odd 2
825.2.c.g.199.2 6 20.3 even 4
825.2.c.g.199.5 6 20.7 even 4
1815.2.a.m.1.1 3 44.43 even 2
2475.2.a.bb.1.3 3 60.59 even 2
2475.2.c.r.199.2 6 60.47 odd 4
2475.2.c.r.199.5 6 60.23 odd 4
2640.2.a.be.1.3 3 1.1 even 1 trivial
5445.2.a.z.1.3 3 132.131 odd 2
7920.2.a.cj.1.3 3 3.2 odd 2
8085.2.a.bk.1.3 3 28.27 even 2
9075.2.a.cf.1.3 3 220.219 even 2