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 473.4
Root \(-1.34002i\) of defining polynomial
Character \(\chi\) \(=\) 520.473
Dual form 520.2.bh.f.177.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.847275 - 0.847275i) q^{3} +(-2.13480 - 0.665310i) q^{5} -3.46256 q^{7} -1.56425i q^{9} +(2.56559 + 2.56559i) q^{11} +(1.26161 + 3.37762i) q^{13} +(1.24506 + 2.37246i) q^{15} +(4.14954 + 4.14954i) q^{17} +(1.52256 + 1.52256i) q^{19} +(2.93375 + 2.93375i) q^{21} +(-5.13045 + 5.13045i) q^{23} +(4.11472 + 2.84061i) q^{25} +(-3.86718 + 3.86718i) q^{27} -7.81295i q^{29} +(-1.69923 + 1.69923i) q^{31} -4.34753i q^{33} +(7.39187 + 2.30368i) q^{35} -0.417441 q^{37} +(1.79285 - 3.93071i) q^{39} +(-3.55127 + 3.55127i) q^{41} +(-1.08328 + 1.08328i) q^{43} +(-1.04071 + 3.33936i) q^{45} -6.66881 q^{47} +4.98935 q^{49} -7.03160i q^{51} +(3.56311 + 3.56311i) q^{53} +(-3.77011 - 7.18394i) q^{55} -2.58006i q^{57} +(-0.813031 + 0.813031i) q^{59} +8.95826 q^{61} +5.41631i q^{63} +(-0.446112 - 8.04991i) q^{65} -6.91443i q^{67} +8.69381 q^{69} +(-11.1047 + 11.1047i) q^{71} +14.7467i q^{73} +(-1.07953 - 5.89308i) q^{75} +(-8.88353 - 8.88353i) q^{77} +0.436982i q^{79} +1.86038 q^{81} +8.46829 q^{83} +(-6.09769 - 11.6192i) q^{85} +(-6.61972 + 6.61972i) q^{87} +(-10.2037 + 10.2037i) q^{89} +(-4.36840 - 11.6952i) q^{91} +2.87943 q^{93} +(-2.23739 - 4.26334i) q^{95} -4.55796i q^{97} +(4.01323 - 4.01323i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{11} + 4 q^{13} + 10 q^{15} + 6 q^{17} + 2 q^{19} + 6 q^{21} - 14 q^{23} - 14 q^{25} + 6 q^{27} + 10 q^{31} - 16 q^{35} + 4 q^{37} - 32 q^{39} + 28 q^{41} + 2 q^{45} - 20 q^{47} + 16 q^{49} + 20 q^{53}+ \cdots + 42 q^{99}+O(q^{100}) \) 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
\(3\) −0.847275 − 0.847275i −0.489175 − 0.489175i 0.418871 − 0.908046i \(-0.362426\pi\)
−0.908046 + 0.418871i \(0.862426\pi\)
\(4\) 0 0
\(5\) −2.13480 − 0.665310i −0.954711 − 0.297536i
\(6\) 0 0
\(7\) −3.46256 −1.30873 −0.654363 − 0.756180i \(-0.727063\pi\)
−0.654363 + 0.756180i \(0.727063\pi\)
\(8\) 0 0
\(9\) − 1.56425i − 0.521416i
\(10\) 0 0
\(11\) 2.56559 + 2.56559i 0.773555 + 0.773555i 0.978726 − 0.205171i \(-0.0657750\pi\)
−0.205171 + 0.978726i \(0.565775\pi\)
\(12\) 0 0
\(13\) 1.26161 + 3.37762i 0.349907 + 0.936784i
\(14\) 0 0
\(15\) 1.24506 + 2.37246i 0.321473 + 0.612567i
\(16\) 0 0
\(17\) 4.14954 + 4.14954i 1.00641 + 1.00641i 0.999979 + 0.00643122i \(0.00204713\pi\)
0.00643122 + 0.999979i \(0.497953\pi\)
\(18\) 0 0
\(19\) 1.52256 + 1.52256i 0.349300 + 0.349300i 0.859849 − 0.510549i \(-0.170558\pi\)
−0.510549 + 0.859849i \(0.670558\pi\)
\(20\) 0 0
\(21\) 2.93375 + 2.93375i 0.640196 + 0.640196i
\(22\) 0 0
\(23\) −5.13045 + 5.13045i −1.06977 + 1.06977i −0.0723967 + 0.997376i \(0.523065\pi\)
−0.997376 + 0.0723967i \(0.976935\pi\)
\(24\) 0 0
\(25\) 4.11472 + 2.84061i 0.822945 + 0.568121i
\(26\) 0 0
\(27\) −3.86718 + 3.86718i −0.744238 + 0.744238i
\(28\) 0 0
\(29\) − 7.81295i − 1.45083i −0.688313 − 0.725414i \(-0.741648\pi\)
0.688313 − 0.725414i \(-0.258352\pi\)
\(30\) 0 0
\(31\) −1.69923 + 1.69923i −0.305191 + 0.305191i −0.843041 − 0.537850i \(-0.819237\pi\)
0.537850 + 0.843041i \(0.319237\pi\)
\(32\) 0 0
\(33\) − 4.34753i − 0.756807i
\(34\) 0 0
\(35\) 7.39187 + 2.30368i 1.24945 + 0.389393i
\(36\) 0 0
\(37\) −0.417441 −0.0686268 −0.0343134 − 0.999411i \(-0.510924\pi\)
−0.0343134 + 0.999411i \(0.510924\pi\)
\(38\) 0 0
\(39\) 1.79285 − 3.93071i 0.287085 − 0.629417i
\(40\) 0 0
\(41\) −3.55127 + 3.55127i −0.554615 + 0.554615i −0.927769 − 0.373154i \(-0.878276\pi\)
0.373154 + 0.927769i \(0.378276\pi\)
\(42\) 0 0
\(43\) −1.08328 + 1.08328i −0.165198 + 0.165198i −0.784865 − 0.619667i \(-0.787268\pi\)
0.619667 + 0.784865i \(0.287268\pi\)
\(44\) 0 0
\(45\) −1.04071 + 3.33936i −0.155140 + 0.497802i
\(46\) 0 0
\(47\) −6.66881 −0.972746 −0.486373 − 0.873751i \(-0.661680\pi\)
−0.486373 + 0.873751i \(0.661680\pi\)
\(48\) 0 0
\(49\) 4.98935 0.712764
\(50\) 0 0
\(51\) − 7.03160i − 0.984621i
\(52\) 0 0
\(53\) 3.56311 + 3.56311i 0.489430 + 0.489430i 0.908126 − 0.418696i \(-0.137513\pi\)
−0.418696 + 0.908126i \(0.637513\pi\)
\(54\) 0 0
\(55\) −3.77011 − 7.18394i −0.508361 − 0.968682i
\(56\) 0 0
\(57\) − 2.58006i − 0.341737i
\(58\) 0 0
\(59\) −0.813031 + 0.813031i −0.105848 + 0.105848i −0.758047 − 0.652200i \(-0.773846\pi\)
0.652200 + 0.758047i \(0.273846\pi\)
\(60\) 0 0
\(61\) 8.95826 1.14699 0.573494 − 0.819210i \(-0.305588\pi\)
0.573494 + 0.819210i \(0.305588\pi\)
\(62\) 0 0
\(63\) 5.41631i 0.682391i
\(64\) 0 0
\(65\) −0.446112 − 8.04991i −0.0553333 − 0.998468i
\(66\) 0 0
\(67\) − 6.91443i − 0.844733i −0.906425 − 0.422366i \(-0.861200\pi\)
0.906425 − 0.422366i \(-0.138800\pi\)
\(68\) 0 0
\(69\) 8.69381 1.04661
\(70\) 0 0
\(71\) −11.1047 + 11.1047i −1.31789 + 1.31789i −0.402441 + 0.915446i \(0.631838\pi\)
−0.915446 + 0.402441i \(0.868162\pi\)
\(72\) 0 0
\(73\) 14.7467i 1.72597i 0.505231 + 0.862984i \(0.331407\pi\)
−0.505231 + 0.862984i \(0.668593\pi\)
\(74\) 0 0
\(75\) −1.07953 − 5.89308i −0.124653 − 0.680474i
\(76\) 0 0
\(77\) −8.88353 − 8.88353i −1.01237 − 1.01237i
\(78\) 0 0
\(79\) 0.436982i 0.0491643i 0.999698 + 0.0245822i \(0.00782554\pi\)
−0.999698 + 0.0245822i \(0.992174\pi\)
\(80\) 0 0
\(81\) 1.86038 0.206709
\(82\) 0 0
\(83\) 8.46829 0.929516 0.464758 − 0.885438i \(-0.346141\pi\)
0.464758 + 0.885438i \(0.346141\pi\)
\(84\) 0 0
\(85\) −6.09769 − 11.6192i −0.661388 − 1.26027i
\(86\) 0 0
\(87\) −6.61972 + 6.61972i −0.709708 + 0.709708i
\(88\) 0 0
\(89\) −10.2037 + 10.2037i −1.08159 + 1.08159i −0.0852254 + 0.996362i \(0.527161\pi\)
−0.996362 + 0.0852254i \(0.972839\pi\)
\(90\) 0 0
\(91\) −4.36840 − 11.6952i −0.457933 − 1.22599i
\(92\) 0 0
\(93\) 2.87943 0.298583
\(94\) 0 0
\(95\) −2.23739 − 4.26334i −0.229551 − 0.437409i
\(96\) 0 0
\(97\) − 4.55796i − 0.462791i −0.972860 − 0.231395i \(-0.925671\pi\)
0.972860 − 0.231395i \(-0.0743291\pi\)
\(98\) 0 0
\(99\) 4.01323 − 4.01323i 0.403344 − 0.403344i
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.473.4 yes 20
4.3 odd 2 1040.2.cd.p.993.7 20
5.2 odd 4 520.2.w.f.57.4 ✓ 20
13.8 odd 4 520.2.w.f.73.4 yes 20
20.7 even 4 1040.2.bg.p.577.7 20
52.47 even 4 1040.2.bg.p.593.7 20
65.47 even 4 inner 520.2.bh.f.177.4 yes 20
260.47 odd 4 1040.2.cd.p.177.7 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.w.f.57.4 ✓ 20 5.2 odd 4
520.2.w.f.73.4 yes 20 13.8 odd 4
520.2.bh.f.177.4 yes 20 65.47 even 4 inner
520.2.bh.f.473.4 yes 20 1.1 even 1 trivial
1040.2.bg.p.577.7 20 20.7 even 4
1040.2.bg.p.593.7 20 52.47 even 4
1040.2.cd.p.177.7 20 260.47 odd 4
1040.2.cd.p.993.7 20 4.3 odd 2