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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 473.7
Root \(0.535565i\) of defining polynomial
Character \(\chi\) \(=\) 520.473
Dual form 520.2.bh.f.177.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.465613 + 0.465613i) q^{3} +(2.14045 - 0.646900i) q^{5} -1.65752 q^{7} -2.56641i q^{9} +(3.24379 + 3.24379i) q^{11} +(3.59409 + 0.287265i) q^{13} +(1.29783 + 0.695416i) q^{15} +(-3.85848 - 3.85848i) q^{17} +(1.74174 + 1.74174i) q^{19} +(-0.771762 - 0.771762i) q^{21} +(3.60230 - 3.60230i) q^{23} +(4.16304 - 2.76931i) q^{25} +(2.59179 - 2.59179i) q^{27} +6.47261i q^{29} +(-2.96557 + 2.96557i) q^{31} +3.02070i q^{33} +(-3.54783 + 1.07225i) q^{35} +1.82596 q^{37} +(1.53970 + 1.80721i) q^{39} +(1.77852 - 1.77852i) q^{41} +(-6.97485 + 6.97485i) q^{43} +(-1.66021 - 5.49327i) q^{45} +2.47361 q^{47} -4.25264 q^{49} -3.59312i q^{51} +(-2.21322 - 2.21322i) q^{53} +(9.04157 + 4.84476i) q^{55} +1.62195i q^{57} +(6.79578 - 6.79578i) q^{59} +5.72072 q^{61} +4.25387i q^{63} +(7.87880 - 1.71014i) q^{65} -4.08730i q^{67} +3.35456 q^{69} +(-7.85170 + 7.85170i) q^{71} -7.51345i q^{73} +(3.22780 + 0.648939i) q^{75} +(-5.37663 - 5.37663i) q^{77} +2.24062i q^{79} -5.28568 q^{81} -12.5979 q^{83} +(-10.7549 - 5.76283i) q^{85} +(-3.01373 + 3.01373i) q^{87} +(-6.77341 + 6.77341i) q^{89} +(-5.95726 - 0.476147i) q^{91} -2.76162 q^{93} +(4.85483 + 2.60137i) q^{95} -7.23300i q^{97} +(8.32488 - 8.32488i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{11} + 4 q^{13} + 10 q^{15} + 6 q^{17} + 2 q^{19} + 6 q^{21} - 14 q^{23} - 14 q^{25} + 6 q^{27} + 10 q^{31} - 16 q^{35} + 4 q^{37} - 32 q^{39} + 28 q^{41} + 2 q^{45} - 20 q^{47} + 16 q^{49} + 20 q^{53}+ \cdots + 42 q^{99}+O(q^{100}) \) 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
\(3\) 0.465613 + 0.465613i 0.268822 + 0.268822i 0.828625 − 0.559803i \(-0.189123\pi\)
−0.559803 + 0.828625i \(0.689123\pi\)
\(4\) 0 0
\(5\) 2.14045 − 0.646900i 0.957238 − 0.289302i
\(6\) 0 0
\(7\) −1.65752 −0.626483 −0.313241 − 0.949674i \(-0.601415\pi\)
−0.313241 + 0.949674i \(0.601415\pi\)
\(8\) 0 0
\(9\) − 2.56641i − 0.855469i
\(10\) 0 0
\(11\) 3.24379 + 3.24379i 0.978039 + 0.978039i 0.999764 − 0.0217252i \(-0.00691590\pi\)
−0.0217252 + 0.999764i \(0.506916\pi\)
\(12\) 0 0
\(13\) 3.59409 + 0.287265i 0.996821 + 0.0796730i
\(14\) 0 0
\(15\) 1.29783 + 0.695416i 0.335097 + 0.179556i
\(16\) 0 0
\(17\) −3.85848 − 3.85848i −0.935818 − 0.935818i 0.0622426 − 0.998061i \(-0.480175\pi\)
−0.998061 + 0.0622426i \(0.980175\pi\)
\(18\) 0 0
\(19\) 1.74174 + 1.74174i 0.399582 + 0.399582i 0.878086 − 0.478504i \(-0.158821\pi\)
−0.478504 + 0.878086i \(0.658821\pi\)
\(20\) 0 0
\(21\) −0.771762 − 0.771762i −0.168412 − 0.168412i
\(22\) 0 0
\(23\) 3.60230 − 3.60230i 0.751132 − 0.751132i −0.223558 − 0.974691i \(-0.571767\pi\)
0.974691 + 0.223558i \(0.0717673\pi\)
\(24\) 0 0
\(25\) 4.16304 − 2.76931i 0.832608 − 0.553862i
\(26\) 0 0
\(27\) 2.59179 − 2.59179i 0.498791 − 0.498791i
\(28\) 0 0
\(29\) 6.47261i 1.20193i 0.799274 + 0.600967i \(0.205218\pi\)
−0.799274 + 0.600967i \(0.794782\pi\)
\(30\) 0 0
\(31\) −2.96557 + 2.96557i −0.532633 + 0.532633i −0.921355 − 0.388722i \(-0.872917\pi\)
0.388722 + 0.921355i \(0.372917\pi\)
\(32\) 0 0
\(33\) 3.02070i 0.525837i
\(34\) 0 0
\(35\) −3.54783 + 1.07225i −0.599693 + 0.181243i
\(36\) 0 0
\(37\) 1.82596 0.300185 0.150093 − 0.988672i \(-0.452043\pi\)
0.150093 + 0.988672i \(0.452043\pi\)
\(38\) 0 0
\(39\) 1.53970 + 1.80721i 0.246550 + 0.289385i
\(40\) 0 0
\(41\) 1.77852 − 1.77852i 0.277758 − 0.277758i −0.554455 − 0.832213i \(-0.687073\pi\)
0.832213 + 0.554455i \(0.187073\pi\)
\(42\) 0 0
\(43\) −6.97485 + 6.97485i −1.06365 + 1.06365i −0.0658237 + 0.997831i \(0.520967\pi\)
−0.997831 + 0.0658237i \(0.979033\pi\)
\(44\) 0 0
\(45\) −1.66021 − 5.49327i −0.247489 − 0.818888i
\(46\) 0 0
\(47\) 2.47361 0.360812 0.180406 − 0.983592i \(-0.442259\pi\)
0.180406 + 0.983592i \(0.442259\pi\)
\(48\) 0 0
\(49\) −4.25264 −0.607520
\(50\) 0 0
\(51\) − 3.59312i − 0.503137i
\(52\) 0 0
\(53\) −2.21322 − 2.21322i −0.304009 − 0.304009i 0.538571 − 0.842580i \(-0.318964\pi\)
−0.842580 + 0.538571i \(0.818964\pi\)
\(54\) 0 0
\(55\) 9.04157 + 4.84476i 1.21916 + 0.653267i
\(56\) 0 0
\(57\) 1.62195i 0.214833i
\(58\) 0 0
\(59\) 6.79578 − 6.79578i 0.884736 − 0.884736i −0.109276 − 0.994011i \(-0.534853\pi\)
0.994011 + 0.109276i \(0.0348531\pi\)
\(60\) 0 0
\(61\) 5.72072 0.732464 0.366232 − 0.930524i \(-0.380648\pi\)
0.366232 + 0.930524i \(0.380648\pi\)
\(62\) 0 0
\(63\) 4.25387i 0.535937i
\(64\) 0 0
\(65\) 7.87880 − 1.71014i 0.977244 − 0.212117i
\(66\) 0 0
\(67\) − 4.08730i − 0.499343i −0.968331 − 0.249672i \(-0.919677\pi\)
0.968331 − 0.249672i \(-0.0803227\pi\)
\(68\) 0 0
\(69\) 3.35456 0.403842
\(70\) 0 0
\(71\) −7.85170 + 7.85170i −0.931825 + 0.931825i −0.997820 − 0.0659945i \(-0.978978\pi\)
0.0659945 + 0.997820i \(0.478978\pi\)
\(72\) 0 0
\(73\) − 7.51345i − 0.879382i −0.898149 − 0.439691i \(-0.855088\pi\)
0.898149 − 0.439691i \(-0.144912\pi\)
\(74\) 0 0
\(75\) 3.22780 + 0.648939i 0.372714 + 0.0749331i
\(76\) 0 0
\(77\) −5.37663 − 5.37663i −0.612724 − 0.612724i
\(78\) 0 0
\(79\) 2.24062i 0.252089i 0.992025 + 0.126045i \(0.0402283\pi\)
−0.992025 + 0.126045i \(0.959772\pi\)
\(80\) 0 0
\(81\) −5.28568 −0.587297
\(82\) 0 0
\(83\) −12.5979 −1.38280 −0.691402 − 0.722471i \(-0.743006\pi\)
−0.691402 + 0.722471i \(0.743006\pi\)
\(84\) 0 0
\(85\) −10.7549 − 5.76283i −1.16654 − 0.625066i
\(86\) 0 0
\(87\) −3.01373 + 3.01373i −0.323106 + 0.323106i
\(88\) 0 0
\(89\) −6.77341 + 6.77341i −0.717980 + 0.717980i −0.968191 − 0.250211i \(-0.919500\pi\)
0.250211 + 0.968191i \(0.419500\pi\)
\(90\) 0 0
\(91\) −5.95726 − 0.476147i −0.624491 − 0.0499138i
\(92\) 0 0
\(93\) −2.76162 −0.286367
\(94\) 0 0
\(95\) 4.85483 + 2.60137i 0.498095 + 0.266895i
\(96\) 0 0
\(97\) − 7.23300i − 0.734400i −0.930142 − 0.367200i \(-0.880316\pi\)
0.930142 − 0.367200i \(-0.119684\pi\)
\(98\) 0 0
\(99\) 8.32488 − 8.32488i 0.836682 − 0.836682i
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.473.7 yes 20
4.3 odd 2 1040.2.cd.p.993.4 20
5.2 odd 4 520.2.w.f.57.7 ✓ 20
13.8 odd 4 520.2.w.f.73.7 yes 20
20.7 even 4 1040.2.bg.p.577.4 20
52.47 even 4 1040.2.bg.p.593.4 20
65.47 even 4 inner 520.2.bh.f.177.7 yes 20
260.47 odd 4 1040.2.cd.p.177.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.w.f.57.7 ✓ 20 5.2 odd 4
520.2.w.f.73.7 yes 20 13.8 odd 4
520.2.bh.f.177.7 yes 20 65.47 even 4 inner
520.2.bh.f.473.7 yes 20 1.1 even 1 trivial
1040.2.bg.p.577.4 20 20.7 even 4
1040.2.bg.p.593.4 20 52.47 even 4
1040.2.cd.p.177.4 20 260.47 odd 4
1040.2.cd.p.993.4 20 4.3 odd 2