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Properties

Label 80.2.n.b.47.1
Level $80$
Weight $2$
Character 80.47
Analytic conductor $0.639$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [80,2,Mod(47,80)] 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("80.47"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.638803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 47.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 80.47
Dual form 80.2.n.b.63.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.73205 - 1.73205i) q^{3} +(-1.00000 - 2.00000i) q^{5} +(1.73205 - 1.73205i) q^{7} +3.00000i q^{9} +3.46410i q^{11} +(1.00000 - 1.00000i) q^{13} +(-1.73205 + 5.19615i) q^{15} +(1.00000 + 1.00000i) q^{17} +6.92820 q^{19} -6.00000 q^{21} +(-1.73205 - 1.73205i) q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{29} -3.46410i q^{31} +(6.00000 - 6.00000i) q^{33} +(-5.19615 - 1.73205i) q^{35} +(5.00000 + 5.00000i) q^{37} -3.46410 q^{39} +2.00000 q^{41} +(-1.73205 - 1.73205i) q^{43} +(6.00000 - 3.00000i) q^{45} +(1.73205 - 1.73205i) q^{47} +1.00000i q^{49} -3.46410i q^{51} +(-7.00000 + 7.00000i) q^{53} +(6.92820 - 3.46410i) q^{55} +(-12.0000 - 12.0000i) q^{57} -6.92820 q^{59} +6.00000 q^{61} +(5.19615 + 5.19615i) q^{63} +(-3.00000 - 1.00000i) q^{65} +(-5.19615 + 5.19615i) q^{67} +6.00000i q^{69} +10.3923i q^{71} +(-7.00000 + 7.00000i) q^{73} +(12.1244 - 1.73205i) q^{75} +(6.00000 + 6.00000i) q^{77} +9.00000 q^{81} +(12.1244 + 12.1244i) q^{83} +(1.00000 - 3.00000i) q^{85} +(-6.92820 + 6.92820i) q^{87} -8.00000i q^{89} -3.46410i q^{91} +(-6.00000 + 6.00000i) q^{93} +(-6.92820 - 13.8564i) q^{95} +(-7.00000 - 7.00000i) q^{97} -10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 4 q^{13} + 4 q^{17} - 24 q^{21} - 12 q^{25} + 24 q^{33} + 20 q^{37} + 8 q^{41} + 24 q^{45} - 28 q^{53} - 48 q^{57} + 24 q^{61} - 12 q^{65} - 28 q^{73} + 24 q^{77} + 36 q^{81} + 4 q^{85}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\)

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
\(3\) −1.73205 − 1.73205i −1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) −1.00000 − 2.00000i −0.447214 − 0.894427i
\(6\) 0 0
\(7\) 1.73205 − 1.73205i 0.654654 − 0.654654i −0.299456 − 0.954110i \(-0.596805\pi\)
0.954110 + 0.299456i \(0.0968053\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 1.00000 − 1.00000i 0.277350 − 0.277350i −0.554700 − 0.832050i \(-0.687167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) −1.73205 + 5.19615i −0.447214 + 1.34164i
\(16\) 0 0
\(17\) 1.00000 + 1.00000i 0.242536 + 0.242536i 0.817898 − 0.575363i \(-0.195139\pi\)
−0.575363 + 0.817898i \(0.695139\pi\)
\(18\) 0 0
\(19\) 6.92820 1.58944 0.794719 − 0.606977i \(-0.207618\pi\)
0.794719 + 0.606977i \(0.207618\pi\)
\(20\) 0 0
\(21\) −6.00000 −1.30931
\(22\) 0 0
\(23\) −1.73205 − 1.73205i −0.361158 − 0.361158i 0.503081 − 0.864239i \(-0.332200\pi\)
−0.864239 + 0.503081i \(0.832200\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.00000i − 0.742781i −0.928477 − 0.371391i \(-0.878881\pi\)
0.928477 − 0.371391i \(-0.121119\pi\)
\(30\) 0 0
\(31\) − 3.46410i − 0.622171i −0.950382 − 0.311086i \(-0.899307\pi\)
0.950382 − 0.311086i \(-0.100693\pi\)
\(32\) 0 0
\(33\) 6.00000 − 6.00000i 1.04447 − 1.04447i
\(34\) 0 0
\(35\) −5.19615 − 1.73205i −0.878310 − 0.292770i
\(36\) 0 0
\(37\) 5.00000 + 5.00000i 0.821995 + 0.821995i 0.986394 − 0.164399i \(-0.0525685\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −3.46410 −0.554700
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −1.73205 − 1.73205i −0.264135 − 0.264135i 0.562596 − 0.826732i \(-0.309803\pi\)
−0.826732 + 0.562596i \(0.809803\pi\)
\(44\) 0 0
\(45\) 6.00000 − 3.00000i 0.894427 − 0.447214i
\(46\) 0 0
\(47\) 1.73205 − 1.73205i 0.252646 − 0.252646i −0.569409 − 0.822054i \(-0.692828\pi\)
0.822054 + 0.569409i \(0.192828\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) − 3.46410i − 0.485071i
\(52\) 0 0
\(53\) −7.00000 + 7.00000i −0.961524 + 0.961524i −0.999287 − 0.0377628i \(-0.987977\pi\)
0.0377628 + 0.999287i \(0.487977\pi\)
\(54\) 0 0
\(55\) 6.92820 − 3.46410i 0.934199 − 0.467099i
\(56\) 0 0
\(57\) −12.0000 − 12.0000i −1.58944 − 1.58944i
\(58\) 0 0
\(59\) −6.92820 −0.901975 −0.450988 − 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 5.19615 + 5.19615i 0.654654 + 0.654654i
\(64\) 0 0
\(65\) −3.00000 − 1.00000i −0.372104 − 0.124035i
\(66\) 0 0
\(67\) −5.19615 + 5.19615i −0.634811 + 0.634811i −0.949271 − 0.314460i \(-0.898177\pi\)
0.314460 + 0.949271i \(0.398177\pi\)
\(68\) 0 0
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) 10.3923i 1.23334i 0.787222 + 0.616670i \(0.211519\pi\)
−0.787222 + 0.616670i \(0.788481\pi\)
\(72\) 0 0
\(73\) −7.00000 + 7.00000i −0.819288 + 0.819288i −0.986005 − 0.166717i \(-0.946683\pi\)
0.166717 + 0.986005i \(0.446683\pi\)
\(74\) 0 0
\(75\) 12.1244 − 1.73205i 1.40000 − 0.200000i
\(76\) 0 0
\(77\) 6.00000 + 6.00000i 0.683763 + 0.683763i
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 12.1244 + 12.1244i 1.33082 + 1.33082i 0.904636 + 0.426185i \(0.140143\pi\)
0.426185 + 0.904636i \(0.359857\pi\)
\(84\) 0 0
\(85\) 1.00000 − 3.00000i 0.108465 − 0.325396i
\(86\) 0 0
\(87\) −6.92820 + 6.92820i −0.742781 + 0.742781i
\(88\) 0 0
\(89\) − 8.00000i − 0.847998i −0.905663 − 0.423999i \(-0.860626\pi\)
0.905663 − 0.423999i \(-0.139374\pi\)
\(90\) 0 0
\(91\) − 3.46410i − 0.363137i
\(92\) 0 0
\(93\) −6.00000 + 6.00000i −0.622171 + 0.622171i
\(94\) 0 0
\(95\) −6.92820 − 13.8564i −0.710819 − 1.42164i
\(96\) 0 0
\(97\) −7.00000 − 7.00000i −0.710742 − 0.710742i 0.255948 − 0.966691i \(-0.417612\pi\)
−0.966691 + 0.255948i \(0.917612\pi\)
\(98\) 0 0
\(99\) −10.3923 −1.04447
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 80.2.n.b.47.1 ✓ 4
3.2 odd 2 720.2.x.d.127.2 4
4.3 odd 2 inner 80.2.n.b.47.2 yes 4
5.2 odd 4 400.2.n.b.143.1 4
5.3 odd 4 inner 80.2.n.b.63.2 yes 4
5.4 even 2 400.2.n.b.207.2 4
8.3 odd 2 320.2.n.i.127.1 4
8.5 even 2 320.2.n.i.127.2 4
12.11 even 2 720.2.x.d.127.1 4
15.2 even 4 3600.2.x.e.2143.2 4
15.8 even 4 720.2.x.d.703.1 4
15.14 odd 2 3600.2.x.e.3007.1 4
16.3 odd 4 1280.2.o.q.127.1 4
16.5 even 4 1280.2.o.r.127.1 4
16.11 odd 4 1280.2.o.r.127.2 4
16.13 even 4 1280.2.o.q.127.2 4
20.3 even 4 inner 80.2.n.b.63.1 yes 4
20.7 even 4 400.2.n.b.143.2 4
20.19 odd 2 400.2.n.b.207.1 4
40.3 even 4 320.2.n.i.63.2 4
40.13 odd 4 320.2.n.i.63.1 4
40.19 odd 2 1600.2.n.r.1407.2 4
40.27 even 4 1600.2.n.r.1343.1 4
40.29 even 2 1600.2.n.r.1407.1 4
40.37 odd 4 1600.2.n.r.1343.2 4
60.23 odd 4 720.2.x.d.703.2 4
60.47 odd 4 3600.2.x.e.2143.1 4
60.59 even 2 3600.2.x.e.3007.2 4
80.3 even 4 1280.2.o.r.383.1 4
80.13 odd 4 1280.2.o.r.383.2 4
80.43 even 4 1280.2.o.q.383.2 4
80.53 odd 4 1280.2.o.q.383.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.n.b.47.1 ✓ 4 1.1 even 1 trivial
80.2.n.b.47.2 yes 4 4.3 odd 2 inner
80.2.n.b.63.1 yes 4 20.3 even 4 inner
80.2.n.b.63.2 yes 4 5.3 odd 4 inner
320.2.n.i.63.1 4 40.13 odd 4
320.2.n.i.63.2 4 40.3 even 4
320.2.n.i.127.1 4 8.3 odd 2
320.2.n.i.127.2 4 8.5 even 2
400.2.n.b.143.1 4 5.2 odd 4
400.2.n.b.143.2 4 20.7 even 4
400.2.n.b.207.1 4 20.19 odd 2
400.2.n.b.207.2 4 5.4 even 2
720.2.x.d.127.1 4 12.11 even 2
720.2.x.d.127.2 4 3.2 odd 2
720.2.x.d.703.1 4 15.8 even 4
720.2.x.d.703.2 4 60.23 odd 4
1280.2.o.q.127.1 4 16.3 odd 4
1280.2.o.q.127.2 4 16.13 even 4
1280.2.o.q.383.1 4 80.53 odd 4
1280.2.o.q.383.2 4 80.43 even 4
1280.2.o.r.127.1 4 16.5 even 4
1280.2.o.r.127.2 4 16.11 odd 4
1280.2.o.r.383.1 4 80.3 even 4
1280.2.o.r.383.2 4 80.13 odd 4
1600.2.n.r.1343.1 4 40.27 even 4
1600.2.n.r.1343.2 4 40.37 odd 4
1600.2.n.r.1407.1 4 40.29 even 2
1600.2.n.r.1407.2 4 40.19 odd 2
3600.2.x.e.2143.1 4 60.47 odd 4
3600.2.x.e.2143.2 4 15.2 even 4
3600.2.x.e.3007.1 4 15.14 odd 2
3600.2.x.e.3007.2 4 60.59 even 2