Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [520,2,Mod(177,520)] 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("520.177"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(520, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.bh (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0] 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: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 41 x^{18} + 640 x^{16} + 4888 x^{14} + 19956 x^{12} + 45364 x^{10} + 57952 x^{8} + 41120 x^{6} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 473.8
Root \(2.54561i\) of defining polynomial
Character \(\chi\) \(=\) 520.473
Dual form 520.2.bh.f.177.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.70333 + 1.70333i) q^{3} +(-2.06961 + 0.846581i) q^{5} +3.32755 q^{7} +2.80266i q^{9} +(1.53449 + 1.53449i) q^{11} +(-0.740837 + 3.52862i) q^{13} +(-4.96724 - 2.08323i) q^{15} +(-2.17503 - 2.17503i) q^{17} +(-0.135956 - 0.135956i) q^{19} +(5.66792 + 5.66792i) q^{21} +(-0.566521 + 0.566521i) q^{23} +(3.56660 - 3.50419i) q^{25} +(0.336128 - 0.336128i) q^{27} +5.39293i q^{29} +(-4.97938 + 4.97938i) q^{31} +5.22750i q^{33} +(-6.88674 + 2.81704i) q^{35} +3.05564 q^{37} +(-7.27229 + 4.74851i) q^{39} +(4.38255 - 4.38255i) q^{41} +(7.49966 - 7.49966i) q^{43} +(-2.37268 - 5.80043i) q^{45} -11.3946 q^{47} +4.07259 q^{49} -7.40960i q^{51} +(-7.33548 - 7.33548i) q^{53} +(-4.47488 - 1.87674i) q^{55} -0.463157i q^{57} +(0.300627 - 0.300627i) q^{59} +13.6424 q^{61} +9.32601i q^{63} +(-1.45402 - 7.93006i) q^{65} +10.2005i q^{67} -1.92994 q^{69} +(4.56710 - 4.56710i) q^{71} -1.21555i q^{73} +(12.0439 + 0.106304i) q^{75} +(5.10611 + 5.10611i) q^{77} -16.2706i q^{79} +9.55307 q^{81} -2.36951 q^{83} +(6.34282 + 2.66014i) q^{85} +(-9.18594 + 9.18594i) q^{87} +(4.78668 - 4.78668i) q^{89} +(-2.46517 + 11.7417i) q^{91} -16.9630 q^{93} +(0.396476 + 0.166279i) q^{95} -1.71744i q^{97} +(-4.30067 + 4.30067i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{11} + 4 q^{13} + 10 q^{15} + 6 q^{17} + 2 q^{19} + 6 q^{21} - 14 q^{23} - 14 q^{25} + 6 q^{27} + 10 q^{31} - 16 q^{35} + 4 q^{37} - 32 q^{39} + 28 q^{41} + 2 q^{45} - 20 q^{47} + 16 q^{49} + 20 q^{53}+ \cdots + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(261\) \(391\) \(417\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 1.70333 + 1.70333i 0.983418 + 0.983418i 0.999865 − 0.0164469i \(-0.00523545\pi\)
−0.0164469 + 0.999865i \(0.505235\pi\)
\(4\) 0 0
\(5\) −2.06961 + 0.846581i −0.925559 + 0.378603i
\(6\) 0 0
\(7\) 3.32755 1.25770 0.628848 − 0.777528i \(-0.283527\pi\)
0.628848 + 0.777528i \(0.283527\pi\)
\(8\) 0 0
\(9\) 2.80266i 0.934221i
\(10\) 0 0
\(11\) 1.53449 + 1.53449i 0.462667 + 0.462667i 0.899529 − 0.436861i \(-0.143910\pi\)
−0.436861 + 0.899529i \(0.643910\pi\)
\(12\) 0 0
\(13\) −0.740837 + 3.52862i −0.205471 + 0.978663i
\(14\) 0 0
\(15\) −4.96724 − 2.08323i −1.28254 − 0.537887i
\(16\) 0 0
\(17\) −2.17503 − 2.17503i −0.527523 − 0.527523i 0.392310 − 0.919833i \(-0.371676\pi\)
−0.919833 + 0.392310i \(0.871676\pi\)
\(18\) 0 0
\(19\) −0.135956 − 0.135956i −0.0311906 − 0.0311906i 0.691340 − 0.722530i \(-0.257021\pi\)
−0.722530 + 0.691340i \(0.757021\pi\)
\(20\) 0 0
\(21\) 5.66792 + 5.66792i 1.23684 + 1.23684i
\(22\) 0 0
\(23\) −0.566521 + 0.566521i −0.118128 + 0.118128i −0.763700 − 0.645572i \(-0.776619\pi\)
0.645572 + 0.763700i \(0.276619\pi\)
\(24\) 0 0
\(25\) 3.56660 − 3.50419i 0.713320 − 0.700838i
\(26\) 0 0
\(27\) 0.336128 − 0.336128i 0.0646879 − 0.0646879i
\(28\) 0 0
\(29\) 5.39293i 1.00144i 0.865609 + 0.500721i \(0.166932\pi\)
−0.865609 + 0.500721i \(0.833068\pi\)
\(30\) 0 0
\(31\) −4.97938 + 4.97938i −0.894323 + 0.894323i −0.994927 − 0.100604i \(-0.967923\pi\)
0.100604 + 0.994927i \(0.467923\pi\)
\(32\) 0 0
\(33\) 5.22750i 0.909991i
\(34\) 0 0
\(35\) −6.88674 + 2.81704i −1.16407 + 0.476167i
\(36\) 0 0
\(37\) 3.05564 0.502344 0.251172 − 0.967943i \(-0.419184\pi\)
0.251172 + 0.967943i \(0.419184\pi\)
\(38\) 0 0
\(39\) −7.27229 + 4.74851i −1.16450 + 0.760371i
\(40\) 0 0
\(41\) 4.38255 − 4.38255i 0.684440 − 0.684440i −0.276558 − 0.960997i \(-0.589194\pi\)
0.960997 + 0.276558i \(0.0891937\pi\)
\(42\) 0 0
\(43\) 7.49966 − 7.49966i 1.14369 − 1.14369i 0.155917 − 0.987770i \(-0.450167\pi\)
0.987770 − 0.155917i \(-0.0498331\pi\)
\(44\) 0 0
\(45\) −2.37268 − 5.80043i −0.353699 − 0.864677i
\(46\) 0 0
\(47\) −11.3946 −1.66207 −0.831037 − 0.556217i \(-0.812252\pi\)
−0.831037 + 0.556217i \(0.812252\pi\)
\(48\) 0 0
\(49\) 4.07259 0.581799
\(50\) 0 0
\(51\) − 7.40960i − 1.03755i
\(52\) 0 0
\(53\) −7.33548 − 7.33548i −1.00761 − 1.00761i −0.999971 − 0.00763424i \(-0.997570\pi\)
−0.00763424 − 0.999971i \(-0.502430\pi\)
\(54\) 0 0
\(55\) −4.47488 − 1.87674i −0.603393 − 0.253059i
\(56\) 0 0
\(57\) − 0.463157i − 0.0613467i
\(58\) 0 0
\(59\) 0.300627 − 0.300627i 0.0391384 − 0.0391384i −0.687267 − 0.726405i \(-0.741190\pi\)
0.726405 + 0.687267i \(0.241190\pi\)
\(60\) 0 0
\(61\) 13.6424 1.74673 0.873365 − 0.487067i \(-0.161933\pi\)
0.873365 + 0.487067i \(0.161933\pi\)
\(62\) 0 0
\(63\) 9.32601i 1.17497i
\(64\) 0 0
\(65\) −1.45402 − 7.93006i −0.180349 − 0.983603i
\(66\) 0 0
\(67\) 10.2005i 1.24619i 0.782145 + 0.623097i \(0.214126\pi\)
−0.782145 + 0.623097i \(0.785874\pi\)
\(68\) 0 0
\(69\) −1.92994 −0.232338
\(70\) 0 0
\(71\) 4.56710 − 4.56710i 0.542014 − 0.542014i −0.382105 − 0.924119i \(-0.624801\pi\)
0.924119 + 0.382105i \(0.124801\pi\)
\(72\) 0 0
\(73\) − 1.21555i − 0.142270i −0.997467 − 0.0711348i \(-0.977338\pi\)
0.997467 − 0.0711348i \(-0.0226620\pi\)
\(74\) 0 0
\(75\) 12.0439 + 0.106304i 1.39071 + 0.0122749i
\(76\) 0 0
\(77\) 5.10611 + 5.10611i 0.581895 + 0.581895i
\(78\) 0 0
\(79\) − 16.2706i − 1.83059i −0.402788 − 0.915293i \(-0.631959\pi\)
0.402788 − 0.915293i \(-0.368041\pi\)
\(80\) 0 0
\(81\) 9.55307 1.06145
\(82\) 0 0
\(83\) −2.36951 −0.260088 −0.130044 − 0.991508i \(-0.541512\pi\)
−0.130044 + 0.991508i \(0.541512\pi\)
\(84\) 0 0
\(85\) 6.34282 + 2.66014i 0.687975 + 0.288532i
\(86\) 0 0
\(87\) −9.18594 + 9.18594i −0.984837 + 0.984837i
\(88\) 0 0
\(89\) 4.78668 − 4.78668i 0.507387 − 0.507387i −0.406337 − 0.913723i \(-0.633194\pi\)
0.913723 + 0.406337i \(0.133194\pi\)
\(90\) 0 0
\(91\) −2.46517 + 11.7417i −0.258420 + 1.23086i
\(92\) 0 0
\(93\) −16.9630 −1.75899
\(94\) 0 0
\(95\) 0.396476 + 0.166279i 0.0406775 + 0.0170599i
\(96\) 0 0
\(97\) − 1.71744i − 0.174379i −0.996192 − 0.0871896i \(-0.972211\pi\)
0.996192 − 0.0871896i \(-0.0277886\pi\)
\(98\) 0 0
\(99\) −4.30067 + 4.30067i −0.432234 + 0.432234i
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 520.2.bh.f.473.8 yes 20
4.3 odd 2 1040.2.cd.p.993.3 20
5.2 odd 4 520.2.w.f.57.8 ✓ 20
13.8 odd 4 520.2.w.f.73.8 yes 20
20.7 even 4 1040.2.bg.p.577.3 20
52.47 even 4 1040.2.bg.p.593.3 20
65.47 even 4 inner 520.2.bh.f.177.8 yes 20
260.47 odd 4 1040.2.cd.p.177.3 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.w.f.57.8 ✓ 20 5.2 odd 4
520.2.w.f.73.8 yes 20 13.8 odd 4
520.2.bh.f.177.8 yes 20 65.47 even 4 inner
520.2.bh.f.473.8 yes 20 1.1 even 1 trivial
1040.2.bg.p.577.3 20 20.7 even 4
1040.2.bg.p.593.3 20 52.47 even 4
1040.2.cd.p.177.3 20 260.47 odd 4
1040.2.cd.p.993.3 20 4.3 odd 2