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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 177.3
Root \(-3.36342i\) of defining polynomial
Character \(\chi\) \(=\) 520.177
Dual form 520.2.bh.f.473.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.27538 + 1.27538i) q^{3} +(0.322673 - 2.21266i) q^{5} -0.873016 q^{7} -0.253208i q^{9} +(-3.02468 + 3.02468i) q^{11} +(0.830975 - 3.50849i) q^{13} +(2.41046 + 3.23353i) q^{15} +(-4.83686 + 4.83686i) q^{17} +(-2.84236 + 2.84236i) q^{19} +(1.11343 - 1.11343i) q^{21} +(-0.0872931 - 0.0872931i) q^{23} +(-4.79176 - 1.42794i) q^{25} +(-3.50321 - 3.50321i) q^{27} +0.0908998i q^{29} +(0.375914 + 0.375914i) q^{31} -7.71526i q^{33} +(-0.281699 + 1.93169i) q^{35} -6.55774 q^{37} +(3.41486 + 5.53448i) q^{39} +(2.70963 + 2.70963i) q^{41} +(-2.61143 - 2.61143i) q^{43} +(-0.560264 - 0.0817035i) q^{45} -11.9505 q^{47} -6.23784 q^{49} -12.3377i q^{51} +(7.68076 - 7.68076i) q^{53} +(5.71662 + 7.66859i) q^{55} -7.25020i q^{57} +(8.58239 + 8.58239i) q^{59} -6.75528 q^{61} +0.221055i q^{63} +(-7.49497 - 2.97076i) q^{65} -7.16908i q^{67} +0.222664 q^{69} +(7.84836 + 7.84836i) q^{71} +8.88473i q^{73} +(7.93250 - 4.29017i) q^{75} +(2.64059 - 2.64059i) q^{77} -0.670370i q^{79} +9.69551 q^{81} +7.14767 q^{83} +(9.14162 + 12.2631i) q^{85} +(-0.115932 - 0.115932i) q^{87} +(7.82988 + 7.82988i) q^{89} +(-0.725455 + 3.06296i) q^{91} -0.958869 q^{93} +(5.37203 + 7.20634i) q^{95} +8.49443i q^{97} +(0.765874 + 0.765874i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{11} + 4 q^{13} + 10 q^{15} + 6 q^{17} + 2 q^{19} + 6 q^{21} - 14 q^{23} - 14 q^{25} + 6 q^{27} + 10 q^{31} - 16 q^{35} + 4 q^{37} - 32 q^{39} + 28 q^{41} + 2 q^{45} - 20 q^{47} + 16 q^{49} + 20 q^{53}+ \cdots + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.27538 + 1.27538i −0.736343 + 0.736343i −0.971868 − 0.235525i \(-0.924319\pi\)
0.235525 + 0.971868i \(0.424319\pi\)
\(4\) 0 0
\(5\) 0.322673 − 2.21266i 0.144304 − 0.989533i
\(6\) 0 0
\(7\) −0.873016 −0.329969 −0.164984 − 0.986296i \(-0.552757\pi\)
−0.164984 + 0.986296i \(0.552757\pi\)
\(8\) 0 0
\(9\) − 0.253208i − 0.0844027i
\(10\) 0 0
\(11\) −3.02468 + 3.02468i −0.911976 + 0.911976i −0.996428 − 0.0844517i \(-0.973086\pi\)
0.0844517 + 0.996428i \(0.473086\pi\)
\(12\) 0 0
\(13\) 0.830975 − 3.50849i 0.230471 − 0.973079i
\(14\) 0 0
\(15\) 2.41046 + 3.23353i 0.622379 + 0.834893i
\(16\) 0 0
\(17\) −4.83686 + 4.83686i −1.17311 + 1.17311i −0.191646 + 0.981464i \(0.561383\pi\)
−0.981464 + 0.191646i \(0.938617\pi\)
\(18\) 0 0
\(19\) −2.84236 + 2.84236i −0.652082 + 0.652082i −0.953494 − 0.301412i \(-0.902542\pi\)
0.301412 + 0.953494i \(0.402542\pi\)
\(20\) 0 0
\(21\) 1.11343 − 1.11343i 0.242970 − 0.242970i
\(22\) 0 0
\(23\) −0.0872931 − 0.0872931i −0.0182019 − 0.0182019i 0.697947 − 0.716149i \(-0.254097\pi\)
−0.716149 + 0.697947i \(0.754097\pi\)
\(24\) 0 0
\(25\) −4.79176 − 1.42794i −0.958353 − 0.285587i
\(26\) 0 0
\(27\) −3.50321 − 3.50321i −0.674194 − 0.674194i
\(28\) 0 0
\(29\) 0.0908998i 0.0168797i 0.999964 + 0.00843984i \(0.00268652\pi\)
−0.999964 + 0.00843984i \(0.997313\pi\)
\(30\) 0 0
\(31\) 0.375914 + 0.375914i 0.0675162 + 0.0675162i 0.740059 − 0.672542i \(-0.234798\pi\)
−0.672542 + 0.740059i \(0.734798\pi\)
\(32\) 0 0
\(33\) − 7.71526i − 1.34305i
\(34\) 0 0
\(35\) −0.281699 + 1.93169i −0.0476158 + 0.326515i
\(36\) 0 0
\(37\) −6.55774 −1.07809 −0.539043 − 0.842278i \(-0.681214\pi\)
−0.539043 + 0.842278i \(0.681214\pi\)
\(38\) 0 0
\(39\) 3.41486 + 5.53448i 0.546814 + 0.886226i
\(40\) 0 0
\(41\) 2.70963 + 2.70963i 0.423174 + 0.423174i 0.886295 − 0.463121i \(-0.153271\pi\)
−0.463121 + 0.886295i \(0.653271\pi\)
\(42\) 0 0
\(43\) −2.61143 − 2.61143i −0.398239 − 0.398239i 0.479373 − 0.877611i \(-0.340864\pi\)
−0.877611 + 0.479373i \(0.840864\pi\)
\(44\) 0 0
\(45\) −0.560264 − 0.0817035i −0.0835193 − 0.0121796i
\(46\) 0 0
\(47\) −11.9505 −1.74316 −0.871582 − 0.490250i \(-0.836906\pi\)
−0.871582 + 0.490250i \(0.836906\pi\)
\(48\) 0 0
\(49\) −6.23784 −0.891120
\(50\) 0 0
\(51\) − 12.3377i − 1.72762i
\(52\) 0 0
\(53\) 7.68076 − 7.68076i 1.05503 − 1.05503i 0.0566387 − 0.998395i \(-0.481962\pi\)
0.998395 − 0.0566387i \(-0.0180383\pi\)
\(54\) 0 0
\(55\) 5.71662 + 7.66859i 0.770829 + 1.03403i
\(56\) 0 0
\(57\) − 7.25020i − 0.960313i
\(58\) 0 0
\(59\) 8.58239 + 8.58239i 1.11733 + 1.11733i 0.992131 + 0.125201i \(0.0399575\pi\)
0.125201 + 0.992131i \(0.460042\pi\)
\(60\) 0 0
\(61\) −6.75528 −0.864925 −0.432462 − 0.901652i \(-0.642355\pi\)
−0.432462 + 0.901652i \(0.642355\pi\)
\(62\) 0 0
\(63\) 0.221055i 0.0278503i
\(64\) 0 0
\(65\) −7.49497 − 2.97076i −0.929636 − 0.368478i
\(66\) 0 0
\(67\) − 7.16908i − 0.875843i −0.899013 − 0.437921i \(-0.855715\pi\)
0.899013 − 0.437921i \(-0.144285\pi\)
\(68\) 0 0
\(69\) 0.222664 0.0268056
\(70\) 0 0
\(71\) 7.84836 + 7.84836i 0.931429 + 0.931429i 0.997795 − 0.0663667i \(-0.0211407\pi\)
−0.0663667 + 0.997795i \(0.521141\pi\)
\(72\) 0 0
\(73\) 8.88473i 1.03988i 0.854203 + 0.519940i \(0.174046\pi\)
−0.854203 + 0.519940i \(0.825954\pi\)
\(74\) 0 0
\(75\) 7.93250 − 4.29017i 0.915967 − 0.495386i
\(76\) 0 0
\(77\) 2.64059 − 2.64059i 0.300924 − 0.300924i
\(78\) 0 0
\(79\) − 0.670370i − 0.0754225i −0.999289 − 0.0377113i \(-0.987993\pi\)
0.999289 − 0.0377113i \(-0.0120067\pi\)
\(80\) 0 0
\(81\) 9.69551 1.07728
\(82\) 0 0
\(83\) 7.14767 0.784559 0.392279 − 0.919846i \(-0.371687\pi\)
0.392279 + 0.919846i \(0.371687\pi\)
\(84\) 0 0
\(85\) 9.14162 + 12.2631i 0.991548 + 1.33012i
\(86\) 0 0
\(87\) −0.115932 − 0.115932i −0.0124292 − 0.0124292i
\(88\) 0 0
\(89\) 7.82988 + 7.82988i 0.829966 + 0.829966i 0.987512 − 0.157546i \(-0.0503581\pi\)
−0.157546 + 0.987512i \(0.550358\pi\)
\(90\) 0 0
\(91\) −0.725455 + 3.06296i −0.0760483 + 0.321086i
\(92\) 0 0
\(93\) −0.958869 −0.0994301
\(94\) 0 0
\(95\) 5.37203 + 7.20634i 0.551159 + 0.739355i
\(96\) 0 0
\(97\) 8.49443i 0.862478i 0.902238 + 0.431239i \(0.141923\pi\)
−0.902238 + 0.431239i \(0.858077\pi\)
\(98\) 0 0
\(99\) 0.765874 + 0.765874i 0.0769732 + 0.0769732i
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.177.3 yes 20
4.3 odd 2 1040.2.cd.p.177.8 20
5.3 odd 4 520.2.w.f.73.3 yes 20
13.5 odd 4 520.2.w.f.57.3 ✓ 20
20.3 even 4 1040.2.bg.p.593.8 20
52.31 even 4 1040.2.bg.p.577.8 20
65.18 even 4 inner 520.2.bh.f.473.3 yes 20
260.83 odd 4 1040.2.cd.p.993.8 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.w.f.57.3 ✓ 20 13.5 odd 4
520.2.w.f.73.3 yes 20 5.3 odd 4
520.2.bh.f.177.3 yes 20 1.1 even 1 trivial
520.2.bh.f.473.3 yes 20 65.18 even 4 inner
1040.2.bg.p.577.8 20 52.31 even 4
1040.2.bg.p.593.8 20 20.3 even 4
1040.2.cd.p.177.8 20 4.3 odd 2
1040.2.cd.p.993.8 20 260.83 odd 4