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Properties

Label 320.2.o.d.223.1
Level $320$
Weight $2$
Character 320.223
Analytic conductor $2.555$
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] = [320,2,Mod(223,320)] 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("320.223"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,2,0,-2] 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: no
Analytic conductor: \(2.55521286468\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 223.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 320.223
Dual form 320.2.o.d.287.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.00000i) q^{3} +(1.00000 - 2.00000i) q^{5} +(-1.00000 - 1.00000i) q^{7} -1.00000i q^{9} +4.00000 q^{11} +(3.00000 - 3.00000i) q^{13} +(3.00000 - 1.00000i) q^{15} +(-3.00000 + 3.00000i) q^{17} +6.00000i q^{19} -2.00000i q^{21} +(-3.00000 + 3.00000i) q^{23} +(-3.00000 - 4.00000i) q^{25} +(4.00000 - 4.00000i) q^{27} -2.00000 q^{29} +6.00000i q^{31} +(4.00000 + 4.00000i) q^{33} +(-3.00000 + 1.00000i) q^{35} +(3.00000 + 3.00000i) q^{37} +6.00000 q^{39} +6.00000 q^{41} +(-3.00000 - 3.00000i) q^{43} +(-2.00000 - 1.00000i) q^{45} +(-9.00000 - 9.00000i) q^{47} -5.00000i q^{49} -6.00000 q^{51} +(-5.00000 + 5.00000i) q^{53} +(4.00000 - 8.00000i) q^{55} +(-6.00000 + 6.00000i) q^{57} -10.0000i q^{59} +12.0000i q^{61} +(-1.00000 + 1.00000i) q^{63} +(-3.00000 - 9.00000i) q^{65} +(-9.00000 + 9.00000i) q^{67} -6.00000 q^{69} +6.00000i q^{71} +(5.00000 + 5.00000i) q^{73} +(1.00000 - 7.00000i) q^{75} +(-4.00000 - 4.00000i) q^{77} +5.00000 q^{81} +(-3.00000 - 3.00000i) q^{83} +(3.00000 + 9.00000i) q^{85} +(-2.00000 - 2.00000i) q^{87} -6.00000 q^{91} +(-6.00000 + 6.00000i) q^{93} +(12.0000 + 6.00000i) q^{95} +(-7.00000 + 7.00000i) q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 8 q^{11} + 6 q^{13} + 6 q^{15} - 6 q^{17} - 6 q^{23} - 6 q^{25} + 8 q^{27} - 4 q^{29} + 8 q^{33} - 6 q^{35} + 6 q^{37} + 12 q^{39} + 12 q^{41} - 6 q^{43} - 4 q^{45} - 18 q^{47}+ \cdots - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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
\(3\) 1.00000 + 1.00000i 0.577350 + 0.577350i 0.934172 − 0.356822i \(-0.116140\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 0 0
\(5\) 1.00000 − 2.00000i 0.447214 − 0.894427i
\(6\) 0 0
\(7\) −1.00000 − 1.00000i −0.377964 − 0.377964i 0.492403 − 0.870367i \(-0.336119\pi\)
−0.870367 + 0.492403i \(0.836119\pi\)
\(8\) 0 0
\(9\) − 1.00000i − 0.333333i
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 3.00000 − 3.00000i 0.832050 − 0.832050i −0.155747 − 0.987797i \(-0.549778\pi\)
0.987797 + 0.155747i \(0.0497784\pi\)
\(14\) 0 0
\(15\) 3.00000 − 1.00000i 0.774597 − 0.258199i
\(16\) 0 0
\(17\) −3.00000 + 3.00000i −0.727607 + 0.727607i −0.970143 − 0.242536i \(-0.922021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) − 2.00000i − 0.436436i
\(22\) 0 0
\(23\) −3.00000 + 3.00000i −0.625543 + 0.625543i −0.946943 − 0.321400i \(-0.895847\pi\)
0.321400 + 0.946943i \(0.395847\pi\)
\(24\) 0 0
\(25\) −3.00000 − 4.00000i −0.600000 − 0.800000i
\(26\) 0 0
\(27\) 4.00000 − 4.00000i 0.769800 − 0.769800i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 6.00000i 1.07763i 0.842424 + 0.538816i \(0.181128\pi\)
−0.842424 + 0.538816i \(0.818872\pi\)
\(32\) 0 0
\(33\) 4.00000 + 4.00000i 0.696311 + 0.696311i
\(34\) 0 0
\(35\) −3.00000 + 1.00000i −0.507093 + 0.169031i
\(36\) 0 0
\(37\) 3.00000 + 3.00000i 0.493197 + 0.493197i 0.909312 − 0.416115i \(-0.136609\pi\)
−0.416115 + 0.909312i \(0.636609\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −3.00000 − 3.00000i −0.457496 − 0.457496i 0.440337 − 0.897833i \(-0.354859\pi\)
−0.897833 + 0.440337i \(0.854859\pi\)
\(44\) 0 0
\(45\) −2.00000 − 1.00000i −0.298142 − 0.149071i
\(46\) 0 0
\(47\) −9.00000 − 9.00000i −1.31278 − 1.31278i −0.919354 − 0.393431i \(-0.871288\pi\)
−0.393431 − 0.919354i \(-0.628712\pi\)
\(48\) 0 0
\(49\) − 5.00000i − 0.714286i
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −5.00000 + 5.00000i −0.686803 + 0.686803i −0.961524 − 0.274721i \(-0.911414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 4.00000 − 8.00000i 0.539360 − 1.07872i
\(56\) 0 0
\(57\) −6.00000 + 6.00000i −0.794719 + 0.794719i
\(58\) 0 0
\(59\) − 10.0000i − 1.30189i −0.759125 − 0.650945i \(-0.774373\pi\)
0.759125 − 0.650945i \(-0.225627\pi\)
\(60\) 0 0
\(61\) 12.0000i 1.53644i 0.640184 + 0.768221i \(0.278858\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −1.00000 + 1.00000i −0.125988 + 0.125988i
\(64\) 0 0
\(65\) −3.00000 − 9.00000i −0.372104 − 1.11631i
\(66\) 0 0
\(67\) −9.00000 + 9.00000i −1.09952 + 1.09952i −0.105059 + 0.994466i \(0.533503\pi\)
−0.994466 + 0.105059i \(0.966497\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 6.00000i 0.712069i 0.934473 + 0.356034i \(0.115871\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(72\) 0 0
\(73\) 5.00000 + 5.00000i 0.585206 + 0.585206i 0.936329 − 0.351123i \(-0.114200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 1.00000 − 7.00000i 0.115470 − 0.808290i
\(76\) 0 0
\(77\) −4.00000 − 4.00000i −0.455842 − 0.455842i
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) −3.00000 − 3.00000i −0.329293 − 0.329293i 0.523025 − 0.852318i \(-0.324804\pi\)
−0.852318 + 0.523025i \(0.824804\pi\)
\(84\) 0 0
\(85\) 3.00000 + 9.00000i 0.325396 + 0.976187i
\(86\) 0 0
\(87\) −2.00000 − 2.00000i −0.214423 − 0.214423i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) −6.00000 + 6.00000i −0.622171 + 0.622171i
\(94\) 0 0
\(95\) 12.0000 + 6.00000i 1.23117 + 0.615587i
\(96\) 0 0
\(97\) −7.00000 + 7.00000i −0.710742 + 0.710742i −0.966691 − 0.255948i \(-0.917612\pi\)
0.255948 + 0.966691i \(0.417612\pi\)
\(98\) 0 0
\(99\) − 4.00000i − 0.402015i
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 320.2.o.d.223.1 yes 2
4.3 odd 2 320.2.o.b.223.1 yes 2
5.2 odd 4 320.2.o.c.287.1 yes 2
5.3 odd 4 1600.2.o.a.607.1 2
5.4 even 2 1600.2.o.b.543.1 2
8.3 odd 2 320.2.o.c.223.1 yes 2
8.5 even 2 320.2.o.a.223.1 ✓ 2
16.3 odd 4 1280.2.n.i.1023.1 2
16.5 even 4 1280.2.n.j.1023.1 2
16.11 odd 4 1280.2.n.d.1023.1 2
16.13 even 4 1280.2.n.c.1023.1 2
20.3 even 4 1600.2.o.d.607.1 2
20.7 even 4 320.2.o.a.287.1 yes 2
20.19 odd 2 1600.2.o.c.543.1 2
40.3 even 4 1600.2.o.b.607.1 2
40.13 odd 4 1600.2.o.c.607.1 2
40.19 odd 2 1600.2.o.a.543.1 2
40.27 even 4 inner 320.2.o.d.287.1 yes 2
40.29 even 2 1600.2.o.d.543.1 2
40.37 odd 4 320.2.o.b.287.1 yes 2
80.27 even 4 1280.2.n.j.767.1 2
80.37 odd 4 1280.2.n.d.767.1 2
80.67 even 4 1280.2.n.c.767.1 2
80.77 odd 4 1280.2.n.i.767.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.o.a.223.1 ✓ 2 8.5 even 2
320.2.o.a.287.1 yes 2 20.7 even 4
320.2.o.b.223.1 yes 2 4.3 odd 2
320.2.o.b.287.1 yes 2 40.37 odd 4
320.2.o.c.223.1 yes 2 8.3 odd 2
320.2.o.c.287.1 yes 2 5.2 odd 4
320.2.o.d.223.1 yes 2 1.1 even 1 trivial
320.2.o.d.287.1 yes 2 40.27 even 4 inner
1280.2.n.c.767.1 2 80.67 even 4
1280.2.n.c.1023.1 2 16.13 even 4
1280.2.n.d.767.1 2 80.37 odd 4
1280.2.n.d.1023.1 2 16.11 odd 4
1280.2.n.i.767.1 2 80.77 odd 4
1280.2.n.i.1023.1 2 16.3 odd 4
1280.2.n.j.767.1 2 80.27 even 4
1280.2.n.j.1023.1 2 16.5 even 4
1600.2.o.a.543.1 2 40.19 odd 2
1600.2.o.a.607.1 2 5.3 odd 4
1600.2.o.b.543.1 2 5.4 even 2
1600.2.o.b.607.1 2 40.3 even 4
1600.2.o.c.543.1 2 20.19 odd 2
1600.2.o.c.607.1 2 40.13 odd 4
1600.2.o.d.543.1 2 40.29 even 2
1600.2.o.d.607.1 2 20.3 even 4