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 177.4
Root \(1.34002i\) of defining polynomial
Character \(\chi\) \(=\) 520.177
Dual form 520.2.bh.f.473.4

$q$-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 \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).

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

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
\(3\) −0.847275 + 0.847275i −0.489175 + 0.489175i −0.908046 − 0.418871i \(-0.862426\pi\)
0.418871 + 0.908046i \(0.362426\pi\)
\(4\) 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 −0.205171 − 0.978726i \(-0.565775\pi\)
0.978726 + 0.205171i \(0.0657750\pi\)
\(12\) 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 0.00643122 − 0.999979i \(-0.497953\pi\)
0.999979 − 0.00643122i \(-0.00204713\pi\)
\(18\) 0 0
\(19\) 1.52256 − 1.52256i 0.349300 − 0.349300i −0.510549 − 0.859849i \(-0.670558\pi\)
0.859849 + 0.510549i \(0.170558\pi\)
\(20\) 0 0
\(21\) 2.93375 − 2.93375i 0.640196 − 0.640196i
\(22\) 0 0
\(23\) −5.13045 − 5.13045i −1.06977 − 1.06977i −0.997376 − 0.0723967i \(-0.976935\pi\)
−0.0723967 − 0.997376i \(-0.523065\pi\)
\(24\) 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 \(0.258352\pi\)
−0.688313 + 0.725414i \(0.741648\pi\)
\(30\) 0 0
\(31\) −1.69923 − 1.69923i −0.305191 − 0.305191i 0.537850 − 0.843041i \(-0.319237\pi\)
−0.843041 + 0.537850i \(0.819237\pi\)
\(32\) 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 0.373154 − 0.927769i \(-0.378276\pi\)
−0.927769 + 0.373154i \(0.878276\pi\)
\(42\) 0 0
\(43\) −1.08328 − 1.08328i −0.165198 − 0.165198i 0.619667 − 0.784865i \(-0.287268\pi\)
−0.784865 + 0.619667i \(0.787268\pi\)
\(44\) 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 −0.418696 − 0.908126i \(-0.637513\pi\)
0.908126 + 0.418696i \(0.137513\pi\)
\(54\) 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 0.652200 − 0.758047i \(-0.273846\pi\)
−0.758047 + 0.652200i \(0.773846\pi\)
\(60\) 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 \(0.138800\pi\)
−0.906425 + 0.422366i \(0.861200\pi\)
\(68\) 0 0
\(69\) 8.69381 1.04661
\(70\) 0 0
\(71\) −11.1047 − 11.1047i −1.31789 − 1.31789i −0.915446 − 0.402441i \(-0.868162\pi\)
−0.402441 − 0.915446i \(-0.631838\pi\)
\(72\) 0 0
\(73\) − 14.7467i − 1.72597i −0.505231 − 0.862984i \(-0.668593\pi\)
0.505231 − 0.862984i \(-0.331407\pi\)
\(74\) 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 \(-0.992174\pi\)
0.999698 − 0.0245822i \(-0.00782554\pi\)
\(80\) 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 −0.996362 − 0.0852254i \(-0.972839\pi\)
−0.0852254 − 0.996362i \(-0.527161\pi\)
\(90\) 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 \(0.0743291\pi\)
−0.972860 + 0.231395i \(0.925671\pi\)
\(98\) 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.177.4 yes 20
4.3 odd 2 1040.2.cd.p.177.7 20
5.3 odd 4 520.2.w.f.73.4 yes 20
13.5 odd 4 520.2.w.f.57.4 ✓ 20
20.3 even 4 1040.2.bg.p.593.7 20
52.31 even 4 1040.2.bg.p.577.7 20
65.18 even 4 inner 520.2.bh.f.473.4 yes 20
260.83 odd 4 1040.2.cd.p.993.7 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.w.f.57.4 ✓ 20 13.5 odd 4
520.2.w.f.73.4 yes 20 5.3 odd 4
520.2.bh.f.177.4 yes 20 1.1 even 1 trivial
520.2.bh.f.473.4 yes 20 65.18 even 4 inner
1040.2.bg.p.577.7 20 52.31 even 4
1040.2.bg.p.593.7 20 20.3 even 4
1040.2.cd.p.177.7 20 4.3 odd 2
1040.2.cd.p.993.7 20 260.83 odd 4