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Properties

Label 1680.2.t.a.1009.1
Level $1680$
Weight $2$
Character 1680.1009
Analytic conductor $13.415$
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] = [1680,2,Mod(1009,1680)] 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("1680.1009"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-4,0,0,0,-2,0,8,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1680.1009
Dual form 1680.2.t.a.1009.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} +6.00000i q^{13} +(-1.00000 + 2.00000i) q^{15} +2.00000i q^{17} +6.00000 q^{19} -1.00000 q^{21} +2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +1.00000i q^{27} -6.00000 q^{29} +2.00000 q^{31} -4.00000i q^{33} +(-1.00000 + 2.00000i) q^{35} -4.00000i q^{37} +6.00000 q^{39} +8.00000 q^{41} -4.00000i q^{43} +(2.00000 + 1.00000i) q^{45} -4.00000i q^{47} -1.00000 q^{49} +2.00000 q^{51} -6.00000i q^{53} +(-8.00000 - 4.00000i) q^{55} -6.00000i q^{57} +4.00000 q^{59} +14.0000 q^{61} +1.00000i q^{63} +(6.00000 - 12.0000i) q^{65} -4.00000i q^{67} +2.00000 q^{69} +10.0000i q^{73} +(4.00000 - 3.00000i) q^{75} -4.00000i q^{77} +1.00000 q^{81} -16.0000i q^{83} +(2.00000 - 4.00000i) q^{85} +6.00000i q^{87} -8.00000 q^{89} +6.00000 q^{91} -2.00000i q^{93} +(-12.0000 - 6.00000i) q^{95} +10.0000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{9} + 8 q^{11} - 2 q^{15} + 12 q^{19} - 2 q^{21} + 6 q^{25} - 12 q^{29} + 4 q^{31} - 2 q^{35} + 12 q^{39} + 16 q^{41} + 4 q^{45} - 2 q^{49} + 4 q^{51} - 16 q^{55} + 8 q^{59} + 28 q^{61}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) −2.00000 − 1.00000i −0.894427 − 0.447214i
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 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
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −1.00000 + 2.00000i −0.258199 + 0.516398i
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) − 4.00000i − 0.696311i
\(34\) 0 0
\(35\) −1.00000 + 2.00000i −0.169031 + 0.338062i
\(36\) 0 0
\(37\) − 4.00000i − 0.657596i −0.944400 − 0.328798i \(-0.893356\pi\)
0.944400 − 0.328798i \(-0.106644\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 2.00000 + 1.00000i 0.298142 + 0.149071i
\(46\) 0 0
\(47\) − 4.00000i − 0.583460i −0.956501 − 0.291730i \(-0.905769\pi\)
0.956501 − 0.291730i \(-0.0942309\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) −8.00000 − 4.00000i −1.07872 − 0.539360i
\(56\) 0 0
\(57\) − 6.00000i − 0.794719i
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 6.00000 − 12.0000i 0.744208 − 1.48842i
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 4.00000 − 3.00000i 0.461880 − 0.346410i
\(76\) 0 0
\(77\) − 4.00000i − 0.455842i
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 16.0000i − 1.75623i −0.478451 − 0.878114i \(-0.658802\pi\)
0.478451 − 0.878114i \(-0.341198\pi\)
\(84\) 0 0
\(85\) 2.00000 − 4.00000i 0.216930 − 0.433861i
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) − 2.00000i − 0.207390i
\(94\) 0 0
\(95\) −12.0000 − 6.00000i −1.23117 − 0.615587i
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 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 1680.2.t.a.1009.1 2
3.2 odd 2 5040.2.t.o.1009.2 2
4.3 odd 2 420.2.k.a.169.2 yes 2
5.2 odd 4 8400.2.a.bh.1.1 1
5.3 odd 4 8400.2.a.cd.1.1 1
5.4 even 2 inner 1680.2.t.a.1009.2 2
12.11 even 2 1260.2.k.d.1009.2 2
15.14 odd 2 5040.2.t.o.1009.1 2
20.3 even 4 2100.2.a.e.1.1 1
20.7 even 4 2100.2.a.j.1.1 1
20.19 odd 2 420.2.k.a.169.1 ✓ 2
28.3 even 6 2940.2.bb.c.1549.1 4
28.11 odd 6 2940.2.bb.h.1549.2 4
28.19 even 6 2940.2.bb.c.949.2 4
28.23 odd 6 2940.2.bb.h.949.1 4
28.27 even 2 2940.2.k.d.589.1 2
60.23 odd 4 6300.2.a.bc.1.1 1
60.47 odd 4 6300.2.a.n.1.1 1
60.59 even 2 1260.2.k.d.1009.1 2
140.19 even 6 2940.2.bb.c.949.1 4
140.39 odd 6 2940.2.bb.h.1549.1 4
140.59 even 6 2940.2.bb.c.1549.2 4
140.79 odd 6 2940.2.bb.h.949.2 4
140.139 even 2 2940.2.k.d.589.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.k.a.169.1 ✓ 2 20.19 odd 2
420.2.k.a.169.2 yes 2 4.3 odd 2
1260.2.k.d.1009.1 2 60.59 even 2
1260.2.k.d.1009.2 2 12.11 even 2
1680.2.t.a.1009.1 2 1.1 even 1 trivial
1680.2.t.a.1009.2 2 5.4 even 2 inner
2100.2.a.e.1.1 1 20.3 even 4
2100.2.a.j.1.1 1 20.7 even 4
2940.2.k.d.589.1 2 28.27 even 2
2940.2.k.d.589.2 2 140.139 even 2
2940.2.bb.c.949.1 4 140.19 even 6
2940.2.bb.c.949.2 4 28.19 even 6
2940.2.bb.c.1549.1 4 28.3 even 6
2940.2.bb.c.1549.2 4 140.59 even 6
2940.2.bb.h.949.1 4 28.23 odd 6
2940.2.bb.h.949.2 4 140.79 odd 6
2940.2.bb.h.1549.1 4 140.39 odd 6
2940.2.bb.h.1549.2 4 28.11 odd 6
5040.2.t.o.1009.1 2 15.14 odd 2
5040.2.t.o.1009.2 2 3.2 odd 2
6300.2.a.n.1.1 1 60.47 odd 4
6300.2.a.bc.1.1 1 60.23 odd 4
8400.2.a.bh.1.1 1 5.2 odd 4
8400.2.a.cd.1.1 1 5.3 odd 4