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Properties

Label 1000.2.bd.a.109.86
Level $1000$
Weight $2$
Character 1000.109
Analytic conductor $7.985$
Analytic rank $0$
Dimension $2960$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1000,2,Mod(29,1000)] 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("1000.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1000, base_ring=CyclotomicField(50)) chi = DirichletCharacter(H, H._module([0, 25, 31])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.bd (of order \(50\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(2960\)
Relative dimension: \(148\) over \(\Q(\zeta_{50})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{50}]$

Embedding invariants

Embedding label 109.86
Character \(\chi\) \(=\) 1000.109
Dual form 1000.2.bd.a.789.86

$q$-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+(0.374237 + 1.36380i) q^{2} +(1.03669 + 1.25314i) q^{3} +(-1.71989 + 1.02077i) q^{4} +(-2.03199 - 0.933291i) q^{5} +(-1.32106 + 1.88280i) q^{6} +(3.03349 - 0.985641i) q^{7} +(-2.03577 - 1.96358i) q^{8} +(0.0665065 - 0.348639i) q^{9} +(0.512377 - 3.12049i) q^{10} +(0.494407 - 3.91363i) q^{11} +(-3.06215 - 1.09705i) q^{12} +(1.16980 - 6.13231i) q^{13} +(2.47946 + 3.76821i) q^{14} +(-0.936990 - 3.51389i) q^{15} +(1.91607 - 3.51122i) q^{16} +(0.0165701 + 0.0645365i) q^{17} +(0.500363 - 0.0397721i) q^{18} +(0.990641 + 0.819530i) q^{19} +(4.44747 - 0.469024i) q^{20} +(4.37992 + 2.77958i) q^{21} +(5.52243 - 0.790354i) q^{22} +(1.62032 - 1.72547i) q^{23} +(0.350185 - 4.58672i) q^{24} +(3.25794 + 3.79287i) q^{25} +(8.80102 - 0.699563i) q^{26} +(4.78143 - 2.62861i) q^{27} +(-4.21117 + 4.79169i) q^{28} +(-9.62483 + 0.605543i) q^{29} +(4.44158 - 2.59289i) q^{30} +(-0.832957 + 0.213867i) q^{31} +(5.50566 + 1.29911i) q^{32} +(5.41687 - 3.43765i) q^{33} +(-0.0818136 + 0.0467503i) q^{34} +(-7.08390 - 0.828320i) q^{35} +(0.241495 + 0.667510i) q^{36} +(-3.43109 - 1.88626i) q^{37} +(-0.746939 + 1.65773i) q^{38} +(8.89736 - 4.89136i) q^{39} +(2.30406 + 5.88993i) q^{40} +(-8.51617 + 7.99722i) q^{41} +(-2.15167 + 7.01356i) q^{42} +(-3.73622 + 2.71452i) q^{43} +(3.14458 + 7.23571i) q^{44} +(-0.460522 + 0.646360i) q^{45} +(2.95958 + 1.56406i) q^{46} +(-2.15495 - 5.44278i) q^{47} +(6.38641 - 1.23894i) q^{48} +(2.56746 - 1.86537i) q^{49} +(-3.95347 + 5.86260i) q^{50} +(-0.0636951 + 0.0876687i) q^{51} +(4.24773 + 11.7410i) q^{52} +(-2.02742 + 3.19470i) q^{53} +(5.37429 + 5.53719i) q^{54} +(-4.65719 + 7.49102i) q^{55} +(-8.11087 - 3.94997i) q^{56} +2.09101i q^{57} +(-4.42781 - 12.8997i) q^{58} +(5.44075 - 2.56022i) q^{59} +(5.19839 + 5.08707i) q^{60} +(10.4519 - 11.1301i) q^{61} +(-0.603394 - 1.05595i) q^{62} +(-0.141886 - 1.12315i) q^{63} +(0.288703 + 7.99479i) q^{64} +(-8.10025 + 11.3690i) q^{65} +(6.71545 + 6.10102i) q^{66} +(0.309930 - 4.92620i) q^{67} +(-0.0943756 - 0.0940816i) q^{68} +(3.84202 + 0.241720i) q^{69} +(-1.52139 - 9.97101i) q^{70} +(14.1537 - 5.60385i) q^{71} +(-0.819973 + 0.579158i) q^{72} +(5.00080 + 2.35320i) q^{73} +(1.28844 - 5.38522i) q^{74} +(-1.37553 + 8.01466i) q^{75} +(-2.54035 - 0.398290i) q^{76} +(-2.35766 - 12.3593i) q^{77} +(10.0006 + 10.3037i) q^{78} +(6.64526 + 8.03273i) q^{79} +(-7.17042 + 5.34650i) q^{80} +(7.26086 + 2.87478i) q^{81} +(-14.0937 - 8.62149i) q^{82} +(-9.27720 + 11.2142i) q^{83} +(-10.3703 - 0.309708i) q^{84} +(0.0265610 - 0.146602i) q^{85} +(-5.10029 - 4.07958i) q^{86} +(-10.7368 - 11.4335i) q^{87} +(-8.69123 + 6.99644i) q^{88} +(-2.42116 + 5.14522i) q^{89} +(-1.05385 - 0.386167i) q^{90} +(-2.49568 - 19.7553i) q^{91} +(-1.02548 + 4.62160i) q^{92} +(-1.13152 - 0.822097i) q^{93} +(6.61640 - 4.97581i) q^{94} +(-1.24811 - 2.58983i) q^{95} +(4.07969 + 8.24612i) q^{96} +(13.3904 - 0.842450i) q^{97} +(3.50483 + 2.80341i) q^{98} +(-1.33156 - 0.432651i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2960 q - 20 q^{2} - 20 q^{4} - 20 q^{6} - 50 q^{7} - 35 q^{8} - 40 q^{9} - 20 q^{10} - 20 q^{12} - 30 q^{14} - 40 q^{15} - 40 q^{17} - 25 q^{18} - 20 q^{20} + 5 q^{22} - 40 q^{23} - 15 q^{24} - 40 q^{25}+ \cdots - 115 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(e\left(\frac{27}{50}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.374237 + 1.36380i 0.264625 + 0.964351i
\(3\) 1.03669 + 1.25314i 0.598531 + 0.723500i 0.979580 − 0.201055i \(-0.0644370\pi\)
−0.381049 + 0.924555i \(0.624437\pi\)
\(4\) −1.71989 + 1.02077i −0.859947 + 0.510384i
\(5\) −2.03199 − 0.933291i −0.908732 − 0.417380i
\(6\) −1.32106 + 1.88280i −0.539322 + 0.768651i
\(7\) 3.03349 − 0.985641i 1.14655 − 0.372537i 0.326709 − 0.945125i \(-0.394060\pi\)
0.819843 + 0.572588i \(0.194060\pi\)
\(8\) −2.03577 − 1.96358i −0.719753 − 0.694231i
\(9\) 0.0665065 − 0.348639i 0.0221688 − 0.116213i
\(10\) 0.512377 − 3.12049i 0.162028 − 0.986786i
\(11\) 0.494407 − 3.91363i 0.149069 − 1.18000i −0.723080 − 0.690765i \(-0.757274\pi\)
0.872149 − 0.489240i \(-0.162726\pi\)
\(12\) −3.06215 − 1.09705i −0.883967 − 0.316691i
\(13\) 1.16980 − 6.13231i 0.324445 − 1.70080i −0.329418 − 0.944184i \(-0.606852\pi\)
0.653862 − 0.756614i \(-0.273148\pi\)
\(14\) 2.47946 + 3.76821i 0.662664 + 1.00710i
\(15\) −0.936990 − 3.51389i −0.241930 − 0.907282i
\(16\) 1.91607 − 3.51122i 0.479017 − 0.877805i
\(17\) 0.0165701 + 0.0645365i 0.00401885 + 0.0156524i 0.970561 − 0.240856i \(-0.0774280\pi\)
−0.966542 + 0.256508i \(0.917428\pi\)
\(18\) 0.500363 − 0.0397721i 0.117937 − 0.00937438i
\(19\) 0.990641 + 0.819530i 0.227269 + 0.188013i 0.744088 − 0.668081i \(-0.232884\pi\)
−0.516820 + 0.856094i \(0.672884\pi\)
\(20\) 4.44747 − 0.469024i 0.994485 − 0.104877i
\(21\) 4.37992 + 2.77958i 0.955778 + 0.606555i
\(22\) 5.52243 − 0.790354i 1.17739 − 0.168504i
\(23\) 1.62032 − 1.72547i 0.337861 − 0.359786i −0.537493 − 0.843268i \(-0.680628\pi\)
0.875354 + 0.483483i \(0.160628\pi\)
\(24\) 0.350185 − 4.58672i 0.0714813 − 0.936259i
\(25\) 3.25794 + 3.79287i 0.651587 + 0.758574i
\(26\) 8.80102 − 0.699563i 1.72602 − 0.137196i
\(27\) 4.78143 − 2.62861i 0.920187 − 0.505877i
\(28\) −4.21117 + 4.79169i −0.795837 + 0.905544i
\(29\) −9.62483 + 0.605543i −1.78729 + 0.112447i −0.921600 − 0.388140i \(-0.873118\pi\)
−0.865687 + 0.500587i \(0.833118\pi\)
\(30\) 4.44158 − 2.59289i 0.810918 − 0.473395i
\(31\) −0.832957 + 0.213867i −0.149603 + 0.0384116i −0.322749 − 0.946485i \(-0.604607\pi\)
0.173145 + 0.984896i \(0.444607\pi\)
\(32\) 5.50566 + 1.29911i 0.973273 + 0.229652i
\(33\) 5.41687 − 3.43765i 0.942956 − 0.598418i
\(34\) −0.0818136 + 0.0467503i −0.0140309 + 0.00801760i
\(35\) −7.08390 − 0.828320i −1.19740 − 0.140012i
\(36\) 0.241495 + 0.667510i 0.0402492 + 0.111252i
\(37\) −3.43109 − 1.88626i −0.564067 − 0.310099i 0.174087 − 0.984730i \(-0.444303\pi\)
−0.738155 + 0.674632i \(0.764303\pi\)
\(38\) −0.746939 + 1.65773i −0.121170 + 0.268920i
\(39\) 8.89736 − 4.89136i 1.42472 − 0.783245i
\(40\) 2.30406 + 5.88993i 0.364304 + 0.931280i
\(41\) −8.51617 + 7.99722i −1.33000 + 1.24896i −0.384586 + 0.923089i \(0.625656\pi\)
−0.945416 + 0.325866i \(0.894344\pi\)
\(42\) −2.15167 + 7.01356i −0.332009 + 1.08222i
\(43\) −3.73622 + 2.71452i −0.569768 + 0.413961i −0.835021 − 0.550218i \(-0.814545\pi\)
0.265253 + 0.964179i \(0.414545\pi\)
\(44\) 3.14458 + 7.23571i 0.474063 + 1.09082i
\(45\) −0.460522 + 0.646360i −0.0686505 + 0.0963537i
\(46\) 2.95958 + 1.56406i 0.436366 + 0.230608i
\(47\) −2.15495 − 5.44278i −0.314332 − 0.793911i −0.997946 − 0.0640673i \(-0.979593\pi\)
0.683614 − 0.729844i \(-0.260407\pi\)
\(48\) 6.38641 − 1.23894i 0.921799 − 0.178825i
\(49\) 2.56746 − 1.86537i 0.366780 − 0.266482i
\(50\) −3.95347 + 5.86260i −0.559105 + 0.829097i
\(51\) −0.0636951 + 0.0876687i −0.00891909 + 0.0122761i
\(52\) 4.24773 + 11.7410i 0.589054 + 1.62819i
\(53\) −2.02742 + 3.19470i −0.278487 + 0.438825i −0.954564 − 0.298006i \(-0.903679\pi\)
0.676077 + 0.736831i \(0.263679\pi\)
\(54\) 5.37429 + 5.53719i 0.731348 + 0.753516i
\(55\) −4.65719 + 7.49102i −0.627975 + 1.01009i
\(56\) −8.11087 − 3.94997i −1.08386 − 0.527837i
\(57\) 2.09101i 0.276961i
\(58\) −4.42781 − 12.8997i −0.581399 − 1.69382i
\(59\) 5.44075 − 2.56022i 0.708326 − 0.333313i −0.0376452 − 0.999291i \(-0.511986\pi\)
0.745971 + 0.665978i \(0.231986\pi\)
\(60\) 5.19839 + 5.08707i 0.671109 + 0.656738i
\(61\) 10.4519 − 11.1301i 1.33823 − 1.42507i 0.515195 − 0.857073i \(-0.327720\pi\)
0.823032 − 0.567994i \(-0.192280\pi\)
\(62\) −0.603394 − 1.05595i −0.0766311 − 0.134106i
\(63\) −0.141886 − 1.12315i −0.0178760 − 0.141503i
\(64\) 0.288703 + 7.99479i 0.0360879 + 0.999349i
\(65\) −8.10025 + 11.3690i −1.00471 + 1.41015i
\(66\) 6.71545 + 6.10102i 0.826615 + 0.750984i
\(67\) 0.309930 − 4.92620i 0.0378640 − 0.601831i −0.932649 − 0.360784i \(-0.882509\pi\)
0.970513 − 0.241047i \(-0.0774908\pi\)
\(68\) −0.0943756 − 0.0940816i −0.0114447 − 0.0114091i
\(69\) 3.84202 + 0.241720i 0.462525 + 0.0290996i
\(70\) −1.52139 − 9.97101i −0.181841 − 1.19176i
\(71\) 14.1537 − 5.60385i 1.67974 − 0.665055i 0.681941 − 0.731407i \(-0.261136\pi\)
0.997796 + 0.0663522i \(0.0211361\pi\)
\(72\) −0.819973 + 0.579158i −0.0966347 + 0.0682544i
\(73\) 5.00080 + 2.35320i 0.585300 + 0.275421i 0.695544 − 0.718483i \(-0.255163\pi\)
−0.110244 + 0.993904i \(0.535163\pi\)
\(74\) 1.28844 − 5.38522i 0.149778 − 0.626019i
\(75\) −1.37553 + 8.01466i −0.158832 + 0.925453i
\(76\) −2.54035 − 0.398290i −0.291398 − 0.0456870i
\(77\) −2.35766 − 12.3593i −0.268680 − 1.40847i
\(78\) 10.0006 + 10.3037i 1.13234 + 1.16666i
\(79\) 6.64526 + 8.03273i 0.747650 + 0.903753i 0.997923 − 0.0644220i \(-0.0205204\pi\)
−0.250273 + 0.968175i \(0.580520\pi\)
\(80\) −7.17042 + 5.34650i −0.801677 + 0.597757i
\(81\) 7.26086 + 2.87478i 0.806762 + 0.319420i
\(82\) −14.0937 − 8.62149i −1.55638 − 0.952084i
\(83\) −9.27720 + 11.2142i −1.01831 + 1.23092i −0.0455797 + 0.998961i \(0.514513\pi\)
−0.972726 + 0.231959i \(0.925487\pi\)
\(84\) −10.3703 − 0.309708i −1.13149 − 0.0337919i
\(85\) 0.0265610 − 0.146602i 0.00288094 − 0.0159012i
\(86\) −5.10029 − 4.07958i −0.549979 − 0.439912i
\(87\) −10.7368 − 11.4335i −1.15110 − 1.22580i
\(88\) −8.69123 + 6.99644i −0.926489 + 0.745823i
\(89\) −2.42116 + 5.14522i −0.256642 + 0.545392i −0.991078 − 0.133286i \(-0.957447\pi\)
0.734435 + 0.678679i \(0.237447\pi\)
\(90\) −1.05385 − 0.386167i −0.111085 − 0.0407056i
\(91\) −2.49568 − 19.7553i −0.261618 − 2.07092i
\(92\) −1.02548 + 4.62160i −0.106914 + 0.481835i
\(93\) −1.13152 − 0.822097i −0.117333 − 0.0852475i
\(94\) 6.61640 − 4.97581i 0.682429 − 0.513215i
\(95\) −1.24811 − 2.58983i −0.128053 − 0.265711i
\(96\) 4.07969 + 8.24612i 0.416381 + 0.841616i
\(97\) 13.3904 − 0.842450i 1.35959 − 0.0855379i 0.633820 − 0.773480i \(-0.281486\pi\)
0.725765 + 0.687942i \(0.241486\pi\)
\(98\) 3.50483 + 2.80341i 0.354041 + 0.283187i
\(99\) −1.33156 − 0.432651i −0.133827 − 0.0434831i
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 1000.2.bd.a.109.86 yes 2960
8.5 even 2 inner 1000.2.bd.a.109.50 ✓ 2960
125.39 even 50 inner 1000.2.bd.a.789.50 yes 2960
1000.789 even 50 inner 1000.2.bd.a.789.86 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.109.50 ✓ 2960 8.5 even 2 inner
1000.2.bd.a.109.86 yes 2960 1.1 even 1 trivial
1000.2.bd.a.789.50 yes 2960 125.39 even 50 inner
1000.2.bd.a.789.86 yes 2960 1000.789 even 50 inner