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Properties

Label 960.3.l.d.641.2
Level $960$
Weight $3$
Character 960.641
Analytic conductor $26.158$
Analytic rank $0$
Dimension $2$
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] = [960,3,Mod(641,960)] 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("960.641"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,4,0,0,0,-4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.2
Root \(2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 960.641
Dual form 960.3.l.d.641.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.00000 + 2.23607i) q^{3} -2.23607i q^{5} -2.00000 q^{7} +(-1.00000 + 8.94427i) q^{9} -13.4164i q^{11} -8.00000 q^{13} +(5.00000 - 4.47214i) q^{15} -13.4164i q^{17} -34.0000 q^{19} +(-4.00000 - 4.47214i) q^{21} +40.2492i q^{23} -5.00000 q^{25} +(-22.0000 + 15.6525i) q^{27} -40.2492i q^{29} -14.0000 q^{31} +(30.0000 - 26.8328i) q^{33} +4.47214i q^{35} -56.0000 q^{37} +(-16.0000 - 17.8885i) q^{39} +26.8328i q^{41} +8.00000 q^{43} +(20.0000 + 2.23607i) q^{45} -40.2492i q^{47} -45.0000 q^{49} +(30.0000 - 26.8328i) q^{51} -40.2492i q^{53} -30.0000 q^{55} +(-68.0000 - 76.0263i) q^{57} -13.4164i q^{59} +46.0000 q^{61} +(2.00000 - 17.8885i) q^{63} +17.8885i q^{65} +32.0000 q^{67} +(-90.0000 + 80.4984i) q^{69} +53.6656i q^{71} -106.000 q^{73} +(-10.0000 - 11.1803i) q^{75} +26.8328i q^{77} +22.0000 q^{79} +(-79.0000 - 17.8885i) q^{81} +120.748i q^{83} -30.0000 q^{85} +(90.0000 - 80.4984i) q^{87} -107.331i q^{89} +16.0000 q^{91} +(-28.0000 - 31.3050i) q^{93} +76.0263i q^{95} +122.000 q^{97} +(120.000 + 13.4164i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 4 q^{7} - 2 q^{9} - 16 q^{13} + 10 q^{15} - 68 q^{19} - 8 q^{21} - 10 q^{25} - 44 q^{27} - 28 q^{31} + 60 q^{33} - 112 q^{37} - 32 q^{39} + 16 q^{43} + 40 q^{45} - 90 q^{49} + 60 q^{51}+ \cdots + 240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 2.00000 + 2.23607i 0.666667 + 0.745356i
\(4\) 0 0
\(5\) − 2.23607i − 0.447214i
\(6\) 0 0
\(7\) −2.00000 −0.285714 −0.142857 − 0.989743i \(-0.545629\pi\)
−0.142857 + 0.989743i \(0.545629\pi\)
\(8\) 0 0
\(9\) −1.00000 + 8.94427i −0.111111 + 0.993808i
\(10\) 0 0
\(11\) − 13.4164i − 1.21967i −0.792527 − 0.609837i \(-0.791235\pi\)
0.792527 − 0.609837i \(-0.208765\pi\)
\(12\) 0 0
\(13\) −8.00000 −0.615385 −0.307692 − 0.951486i \(-0.599557\pi\)
−0.307692 + 0.951486i \(0.599557\pi\)
\(14\) 0 0
\(15\) 5.00000 − 4.47214i 0.333333 − 0.298142i
\(16\) 0 0
\(17\) − 13.4164i − 0.789200i −0.918853 − 0.394600i \(-0.870883\pi\)
0.918853 − 0.394600i \(-0.129117\pi\)
\(18\) 0 0
\(19\) −34.0000 −1.78947 −0.894737 − 0.446594i \(-0.852637\pi\)
−0.894737 + 0.446594i \(0.852637\pi\)
\(20\) 0 0
\(21\) −4.00000 − 4.47214i −0.190476 − 0.212959i
\(22\) 0 0
\(23\) 40.2492i 1.74997i 0.484153 + 0.874983i \(0.339128\pi\)
−0.484153 + 0.874983i \(0.660872\pi\)
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −22.0000 + 15.6525i −0.814815 + 0.579721i
\(28\) 0 0
\(29\) − 40.2492i − 1.38790i −0.720021 − 0.693952i \(-0.755868\pi\)
0.720021 − 0.693952i \(-0.244132\pi\)
\(30\) 0 0
\(31\) −14.0000 −0.451613 −0.225806 − 0.974172i \(-0.572502\pi\)
−0.225806 + 0.974172i \(0.572502\pi\)
\(32\) 0 0
\(33\) 30.0000 − 26.8328i 0.909091 − 0.813116i
\(34\) 0 0
\(35\) 4.47214i 0.127775i
\(36\) 0 0
\(37\) −56.0000 −1.51351 −0.756757 − 0.653697i \(-0.773217\pi\)
−0.756757 + 0.653697i \(0.773217\pi\)
\(38\) 0 0
\(39\) −16.0000 − 17.8885i −0.410256 − 0.458681i
\(40\) 0 0
\(41\) 26.8328i 0.654459i 0.944945 + 0.327229i \(0.106115\pi\)
−0.944945 + 0.327229i \(0.893885\pi\)
\(42\) 0 0
\(43\) 8.00000 0.186047 0.0930233 − 0.995664i \(-0.470347\pi\)
0.0930233 + 0.995664i \(0.470347\pi\)
\(44\) 0 0
\(45\) 20.0000 + 2.23607i 0.444444 + 0.0496904i
\(46\) 0 0
\(47\) − 40.2492i − 0.856366i −0.903692 − 0.428183i \(-0.859154\pi\)
0.903692 − 0.428183i \(-0.140846\pi\)
\(48\) 0 0
\(49\) −45.0000 −0.918367
\(50\) 0 0
\(51\) 30.0000 − 26.8328i 0.588235 − 0.526134i
\(52\) 0 0
\(53\) − 40.2492i − 0.759419i −0.925106 − 0.379710i \(-0.876024\pi\)
0.925106 − 0.379710i \(-0.123976\pi\)
\(54\) 0 0
\(55\) −30.0000 −0.545455
\(56\) 0 0
\(57\) −68.0000 − 76.0263i −1.19298 − 1.33379i
\(58\) 0 0
\(59\) − 13.4164i − 0.227397i −0.993515 − 0.113698i \(-0.963730\pi\)
0.993515 − 0.113698i \(-0.0362697\pi\)
\(60\) 0 0
\(61\) 46.0000 0.754098 0.377049 − 0.926193i \(-0.376939\pi\)
0.377049 + 0.926193i \(0.376939\pi\)
\(62\) 0 0
\(63\) 2.00000 − 17.8885i 0.0317460 − 0.283945i
\(64\) 0 0
\(65\) 17.8885i 0.275208i
\(66\) 0 0
\(67\) 32.0000 0.477612 0.238806 − 0.971067i \(-0.423244\pi\)
0.238806 + 0.971067i \(0.423244\pi\)
\(68\) 0 0
\(69\) −90.0000 + 80.4984i −1.30435 + 1.16664i
\(70\) 0 0
\(71\) 53.6656i 0.755854i 0.925835 + 0.377927i \(0.123363\pi\)
−0.925835 + 0.377927i \(0.876637\pi\)
\(72\) 0 0
\(73\) −106.000 −1.45205 −0.726027 − 0.687666i \(-0.758635\pi\)
−0.726027 + 0.687666i \(0.758635\pi\)
\(74\) 0 0
\(75\) −10.0000 − 11.1803i −0.133333 − 0.149071i
\(76\) 0 0
\(77\) 26.8328i 0.348478i
\(78\) 0 0
\(79\) 22.0000 0.278481 0.139241 − 0.990259i \(-0.455534\pi\)
0.139241 + 0.990259i \(0.455534\pi\)
\(80\) 0 0
\(81\) −79.0000 − 17.8885i −0.975309 − 0.220846i
\(82\) 0 0
\(83\) 120.748i 1.45479i 0.686218 + 0.727396i \(0.259269\pi\)
−0.686218 + 0.727396i \(0.740731\pi\)
\(84\) 0 0
\(85\) −30.0000 −0.352941
\(86\) 0 0
\(87\) 90.0000 − 80.4984i 1.03448 − 0.925270i
\(88\) 0 0
\(89\) − 107.331i − 1.20597i −0.797753 − 0.602985i \(-0.793978\pi\)
0.797753 − 0.602985i \(-0.206022\pi\)
\(90\) 0 0
\(91\) 16.0000 0.175824
\(92\) 0 0
\(93\) −28.0000 − 31.3050i −0.301075 − 0.336612i
\(94\) 0 0
\(95\) 76.0263i 0.800277i
\(96\) 0 0
\(97\) 122.000 1.25773 0.628866 − 0.777514i \(-0.283519\pi\)
0.628866 + 0.777514i \(0.283519\pi\)
\(98\) 0 0
\(99\) 120.000 + 13.4164i 1.21212 + 0.135519i
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 960.3.l.d.641.2 2
3.2 odd 2 inner 960.3.l.d.641.1 2
4.3 odd 2 960.3.l.a.641.1 2
8.3 odd 2 60.3.g.a.41.2 yes 2
8.5 even 2 240.3.l.a.161.1 2
12.11 even 2 960.3.l.a.641.2 2
24.5 odd 2 240.3.l.a.161.2 2
24.11 even 2 60.3.g.a.41.1 ✓ 2
40.3 even 4 300.3.b.c.149.2 4
40.13 odd 4 1200.3.c.e.449.3 4
40.19 odd 2 300.3.g.d.101.1 2
40.27 even 4 300.3.b.c.149.3 4
40.29 even 2 1200.3.l.r.401.2 2
40.37 odd 4 1200.3.c.e.449.2 4
72.11 even 6 1620.3.o.b.701.2 4
72.43 odd 6 1620.3.o.b.701.1 4
72.59 even 6 1620.3.o.b.1241.1 4
72.67 odd 6 1620.3.o.b.1241.2 4
120.29 odd 2 1200.3.l.r.401.1 2
120.53 even 4 1200.3.c.e.449.1 4
120.59 even 2 300.3.g.d.101.2 2
120.77 even 4 1200.3.c.e.449.4 4
120.83 odd 4 300.3.b.c.149.4 4
120.107 odd 4 300.3.b.c.149.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.3.g.a.41.1 ✓ 2 24.11 even 2
60.3.g.a.41.2 yes 2 8.3 odd 2
240.3.l.a.161.1 2 8.5 even 2
240.3.l.a.161.2 2 24.5 odd 2
300.3.b.c.149.1 4 120.107 odd 4
300.3.b.c.149.2 4 40.3 even 4
300.3.b.c.149.3 4 40.27 even 4
300.3.b.c.149.4 4 120.83 odd 4
300.3.g.d.101.1 2 40.19 odd 2
300.3.g.d.101.2 2 120.59 even 2
960.3.l.a.641.1 2 4.3 odd 2
960.3.l.a.641.2 2 12.11 even 2
960.3.l.d.641.1 2 3.2 odd 2 inner
960.3.l.d.641.2 2 1.1 even 1 trivial
1200.3.c.e.449.1 4 120.53 even 4
1200.3.c.e.449.2 4 40.37 odd 4
1200.3.c.e.449.3 4 40.13 odd 4
1200.3.c.e.449.4 4 120.77 even 4
1200.3.l.r.401.1 2 120.29 odd 2
1200.3.l.r.401.2 2 40.29 even 2
1620.3.o.b.701.1 4 72.43 odd 6
1620.3.o.b.701.2 4 72.11 even 6
1620.3.o.b.1241.1 4 72.59 even 6
1620.3.o.b.1241.2 4 72.67 odd 6