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(49,320)] 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("320.49"); 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([0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.q (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.534694406811304329216.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 209.8
Root \(0.238945 - 1.39388i\) of defining polynomial
Character \(\chi\) \(=\) 320.209
Dual form 320.2.q.c.49.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.99154 - 1.99154i) q^{3} +(-0.569800 + 2.16225i) q^{5} +1.09033 q^{7} -4.93244i q^{9} +(2.33225 - 2.33225i) q^{11} +(1.80775 - 1.80775i) q^{13} +(3.17142 + 5.44098i) q^{15} +4.93886i q^{17} +(-2.03957 - 2.03957i) q^{19} +(2.17142 - 2.17142i) q^{21} -1.45791 q^{23} +(-4.35066 - 2.46410i) q^{25} +(-3.84853 - 3.84853i) q^{27} +(-0.707323 - 0.707323i) q^{29} -10.1286 q^{31} -9.28951i q^{33} +(-0.621268 + 2.35756i) q^{35} +(4.35066 + 4.35066i) q^{37} -7.20039i q^{39} +10.2753i q^{41} +(2.22457 + 2.22457i) q^{43} +(10.6652 + 2.81051i) q^{45} +2.09458i q^{47} -5.81119 q^{49} +(9.83592 + 9.83592i) q^{51} +(0.215297 + 0.215297i) q^{53} +(3.71399 + 6.37182i) q^{55} -8.12376 q^{57} +(1.16082 - 1.16082i) q^{59} +(-3.46410 - 3.46410i) q^{61} -5.37797i q^{63} +(2.87875 + 4.93886i) q^{65} +(5.04189 - 5.04189i) q^{67} +(-2.90348 + 2.90348i) q^{69} +6.40078i q^{71} +5.24343 q^{73} +(-13.5718 + 3.75714i) q^{75} +(2.54291 - 2.54291i) q^{77} -2.61504 q^{79} -0.531659 q^{81} +(-5.67856 + 5.67856i) q^{83} +(-10.6790 - 2.81416i) q^{85} -2.81732 q^{87} -6.87875i q^{89} +(1.97103 - 1.97103i) q^{91} +(-20.1715 + 20.1715i) q^{93} +(5.57221 - 3.24791i) q^{95} -3.77310i q^{97} +(-11.5037 - 11.5037i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{5} - 8 q^{11} + 8 q^{19} - 16 q^{21} - 16 q^{29} - 16 q^{31} + 24 q^{35} + 8 q^{45} + 16 q^{49} + 16 q^{51} + 24 q^{59} + 32 q^{69} - 48 q^{75} - 16 q^{79} - 16 q^{81} + 16 q^{91} - 32 q^{95}+ \cdots - 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.99154 − 1.99154i 1.14981 − 1.14981i 0.163226 − 0.986589i \(-0.447810\pi\)
0.986589 − 0.163226i \(-0.0521899\pi\)
\(4\) 0 0
\(5\) −0.569800 + 2.16225i −0.254822 + 0.966988i
\(6\) 0 0
\(7\) 1.09033 0.412104 0.206052 − 0.978541i \(-0.433938\pi\)
0.206052 + 0.978541i \(0.433938\pi\)
\(8\) 0 0
\(9\) − 4.93244i − 1.64415i
\(10\) 0 0
\(11\) 2.33225 − 2.33225i 0.703199 − 0.703199i −0.261897 − 0.965096i \(-0.584348\pi\)
0.965096 + 0.261897i \(0.0843481\pi\)
\(12\) 0 0
\(13\) 1.80775 − 1.80775i 0.501379 − 0.501379i −0.410487 − 0.911866i \(-0.634641\pi\)
0.911866 + 0.410487i \(0.134641\pi\)
\(14\) 0 0
\(15\) 3.17142 + 5.44098i 0.818858 + 1.40486i
\(16\) 0 0
\(17\) 4.93886i 1.19785i 0.800806 + 0.598924i \(0.204405\pi\)
−0.800806 + 0.598924i \(0.795595\pi\)
\(18\) 0 0
\(19\) −2.03957 − 2.03957i −0.467909 − 0.467909i 0.433327 − 0.901237i \(-0.357339\pi\)
−0.901237 + 0.433327i \(0.857339\pi\)
\(20\) 0 0
\(21\) 2.17142 − 2.17142i 0.473844 − 0.473844i
\(22\) 0 0
\(23\) −1.45791 −0.303995 −0.151997 − 0.988381i \(-0.548570\pi\)
−0.151997 + 0.988381i \(0.548570\pi\)
\(24\) 0 0
\(25\) −4.35066 − 2.46410i −0.870131 − 0.492820i
\(26\) 0 0
\(27\) −3.84853 − 3.84853i −0.740650 − 0.740650i
\(28\) 0 0
\(29\) −0.707323 − 0.707323i −0.131347 − 0.131347i 0.638377 − 0.769724i \(-0.279606\pi\)
−0.769724 + 0.638377i \(0.779606\pi\)
\(30\) 0 0
\(31\) −10.1286 −1.81915 −0.909575 − 0.415541i \(-0.863592\pi\)
−0.909575 + 0.415541i \(0.863592\pi\)
\(32\) 0 0
\(33\) − 9.28951i − 1.61710i
\(34\) 0 0
\(35\) −0.621268 + 2.35756i −0.105013 + 0.398500i
\(36\) 0 0
\(37\) 4.35066 + 4.35066i 0.715243 + 0.715243i 0.967627 − 0.252384i \(-0.0812146\pi\)
−0.252384 + 0.967627i \(0.581215\pi\)
\(38\) 0 0
\(39\) − 7.20039i − 1.15299i
\(40\) 0 0
\(41\) 10.2753i 1.60473i 0.596833 + 0.802365i \(0.296425\pi\)
−0.596833 + 0.802365i \(0.703575\pi\)
\(42\) 0 0
\(43\) 2.22457 + 2.22457i 0.339244 + 0.339244i 0.856083 − 0.516839i \(-0.172891\pi\)
−0.516839 + 0.856083i \(0.672891\pi\)
\(44\) 0 0
\(45\) 10.6652 + 2.81051i 1.58987 + 0.418966i
\(46\) 0 0
\(47\) 2.09458i 0.305525i 0.988263 + 0.152763i \(0.0488170\pi\)
−0.988263 + 0.152763i \(0.951183\pi\)
\(48\) 0 0
\(49\) −5.81119 −0.830170
\(50\) 0 0
\(51\) 9.83592 + 9.83592i 1.37730 + 1.37730i
\(52\) 0 0
\(53\) 0.215297 + 0.215297i 0.0295733 + 0.0295733i 0.721739 − 0.692166i \(-0.243343\pi\)
−0.692166 + 0.721739i \(0.743343\pi\)
\(54\) 0 0
\(55\) 3.71399 + 6.37182i 0.500794 + 0.859175i
\(56\) 0 0
\(57\) −8.12376 −1.07602
\(58\) 0 0
\(59\) 1.16082 − 1.16082i 0.151126 − 0.151126i −0.627495 − 0.778621i \(-0.715920\pi\)
0.778621 + 0.627495i \(0.215920\pi\)
\(60\) 0 0
\(61\) −3.46410 − 3.46410i −0.443533 − 0.443533i 0.449665 − 0.893197i \(-0.351543\pi\)
−0.893197 + 0.449665i \(0.851543\pi\)
\(62\) 0 0
\(63\) − 5.37797i − 0.677560i
\(64\) 0 0
\(65\) 2.87875 + 4.93886i 0.357065 + 0.612590i
\(66\) 0 0
\(67\) 5.04189 − 5.04189i 0.615965 − 0.615965i −0.328529 − 0.944494i \(-0.606553\pi\)
0.944494 + 0.328529i \(0.106553\pi\)
\(68\) 0 0
\(69\) −2.90348 + 2.90348i −0.349537 + 0.349537i
\(70\) 0 0
\(71\) 6.40078i 0.759633i 0.925062 + 0.379817i \(0.124013\pi\)
−0.925062 + 0.379817i \(0.875987\pi\)
\(72\) 0 0
\(73\) 5.24343 0.613696 0.306848 − 0.951758i \(-0.400726\pi\)
0.306848 + 0.951758i \(0.400726\pi\)
\(74\) 0 0
\(75\) −13.5718 + 3.75714i −1.56714 + 0.433837i
\(76\) 0 0
\(77\) 2.54291 − 2.54291i 0.289791 − 0.289791i
\(78\) 0 0
\(79\) −2.61504 −0.294215 −0.147107 − 0.989121i \(-0.546996\pi\)
−0.147107 + 0.989121i \(0.546996\pi\)
\(80\) 0 0
\(81\) −0.531659 −0.0590733
\(82\) 0 0
\(83\) −5.67856 + 5.67856i −0.623303 + 0.623303i −0.946375 − 0.323071i \(-0.895285\pi\)
0.323071 + 0.946375i \(0.395285\pi\)
\(84\) 0 0
\(85\) −10.6790 − 2.81416i −1.15831 − 0.305239i
\(86\) 0 0
\(87\) −2.81732 −0.302048
\(88\) 0 0
\(89\) − 6.87875i − 0.729146i −0.931175 − 0.364573i \(-0.881215\pi\)
0.931175 − 0.364573i \(-0.118785\pi\)
\(90\) 0 0
\(91\) 1.97103 − 1.97103i 0.206620 − 0.206620i
\(92\) 0 0
\(93\) −20.1715 + 20.1715i −2.09168 + 2.09168i
\(94\) 0 0
\(95\) 5.57221 − 3.24791i 0.571696 − 0.333229i
\(96\) 0 0
\(97\) − 3.77310i − 0.383100i −0.981483 − 0.191550i \(-0.938649\pi\)
0.981483 − 0.191550i \(-0.0613515\pi\)
\(98\) 0 0
\(99\) −11.5037 − 11.5037i −1.15616 − 1.15616i
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.q.c.209.8 16
4.3 odd 2 80.2.q.c.29.3 ✓ 16
5.2 odd 4 1600.2.l.h.401.1 16
5.3 odd 4 1600.2.l.h.401.8 16
5.4 even 2 inner 320.2.q.c.209.1 16
8.3 odd 2 640.2.q.e.289.8 16
8.5 even 2 640.2.q.f.289.1 16
12.11 even 2 720.2.bm.f.109.6 16
16.3 odd 4 640.2.q.e.609.1 16
16.5 even 4 inner 320.2.q.c.49.1 16
16.11 odd 4 80.2.q.c.69.6 yes 16
16.13 even 4 640.2.q.f.609.8 16
20.3 even 4 400.2.l.i.301.2 16
20.7 even 4 400.2.l.i.301.7 16
20.19 odd 2 80.2.q.c.29.6 yes 16
40.19 odd 2 640.2.q.e.289.1 16
40.29 even 2 640.2.q.f.289.8 16
48.11 even 4 720.2.bm.f.469.3 16
60.59 even 2 720.2.bm.f.109.3 16
80.19 odd 4 640.2.q.e.609.8 16
80.27 even 4 400.2.l.i.101.7 16
80.29 even 4 640.2.q.f.609.1 16
80.37 odd 4 1600.2.l.h.1201.1 16
80.43 even 4 400.2.l.i.101.2 16
80.53 odd 4 1600.2.l.h.1201.8 16
80.59 odd 4 80.2.q.c.69.3 yes 16
80.69 even 4 inner 320.2.q.c.49.8 16
240.59 even 4 720.2.bm.f.469.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.q.c.29.3 ✓ 16 4.3 odd 2
80.2.q.c.29.6 yes 16 20.19 odd 2
80.2.q.c.69.3 yes 16 80.59 odd 4
80.2.q.c.69.6 yes 16 16.11 odd 4
320.2.q.c.49.1 16 16.5 even 4 inner
320.2.q.c.49.8 16 80.69 even 4 inner
320.2.q.c.209.1 16 5.4 even 2 inner
320.2.q.c.209.8 16 1.1 even 1 trivial
400.2.l.i.101.2 16 80.43 even 4
400.2.l.i.101.7 16 80.27 even 4
400.2.l.i.301.2 16 20.3 even 4
400.2.l.i.301.7 16 20.7 even 4
640.2.q.e.289.1 16 40.19 odd 2
640.2.q.e.289.8 16 8.3 odd 2
640.2.q.e.609.1 16 16.3 odd 4
640.2.q.e.609.8 16 80.19 odd 4
640.2.q.f.289.1 16 8.5 even 2
640.2.q.f.289.8 16 40.29 even 2
640.2.q.f.609.1 16 80.29 even 4
640.2.q.f.609.8 16 16.13 even 4
720.2.bm.f.109.3 16 60.59 even 2
720.2.bm.f.109.6 16 12.11 even 2
720.2.bm.f.469.3 16 48.11 even 4
720.2.bm.f.469.6 16 240.59 even 4
1600.2.l.h.401.1 16 5.2 odd 4
1600.2.l.h.401.8 16 5.3 odd 4
1600.2.l.h.1201.1 16 80.37 odd 4
1600.2.l.h.1201.8 16 80.53 odd 4