↑

Properties

Label 1000.2.bd.a.789.50
Level $1000$
Weight $2$
Character 1000.789
Analytic conductor $7.985$
Analytic rank $0$
Dimension $2960$
Inner twists $4$

Related objects

Downloads

Learn more

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 789.50
Character \(\chi\) \(=\) 1000.789
Dual form 1000.2.bd.a.109.50

$q$-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+(-0.701642 - 1.22788i) q^{2} +(-1.03669 + 1.25314i) q^{3} +(-1.01540 + 1.72307i) q^{4} +(2.03199 - 0.933291i) q^{5} +(2.26609 + 0.393675i) q^{6} +(3.03349 + 0.985641i) q^{7} +(2.82817 + 0.0378094i) q^{8} +(0.0665065 + 0.348639i) q^{9} +(-2.57170 - 1.84021i) q^{10} +(-0.494407 - 3.91363i) q^{11} +(-1.10660 - 3.05872i) q^{12} +(-1.16980 - 6.13231i) q^{13} +(-0.918173 - 4.41634i) q^{14} +(-0.936990 + 3.51389i) q^{15} +(-1.93794 - 3.49920i) q^{16} +(0.0165701 - 0.0645365i) q^{17} +(0.381424 - 0.326282i) q^{18} +(-0.990641 + 0.819530i) q^{19} +(-0.455146 + 4.44891i) q^{20} +(-4.37992 + 2.77958i) q^{21} +(-4.45859 + 3.35305i) q^{22} +(1.62032 + 1.72547i) q^{23} +(-2.97931 + 3.50490i) q^{24} +(3.25794 - 3.79287i) q^{25} +(-6.70899 + 5.73907i) q^{26} +(-4.78143 - 2.62861i) q^{27} +(-4.77852 + 4.22610i) q^{28} +(9.62483 + 0.605543i) q^{29} +(4.97208 - 1.31498i) q^{30} +(-0.832957 - 0.213867i) q^{31} +(-2.93687 + 4.83475i) q^{32} +(5.41687 + 3.43765i) q^{33} +(-0.0908696 + 0.0249353i) q^{34} +(7.08390 - 0.828320i) q^{35} +(-0.668260 - 0.239411i) q^{36} +(3.43109 - 1.88626i) q^{37} +(1.70136 + 0.641376i) q^{38} +(8.89736 + 4.89136i) q^{39} +(5.78210 - 2.56268i) q^{40} +(-8.51617 - 7.99722i) q^{41} +(6.48615 + 3.42776i) q^{42} +(3.73622 + 2.71452i) q^{43} +(7.24548 + 3.12199i) q^{44} +(0.460522 + 0.646360i) q^{45} +(0.981789 - 3.20023i) q^{46} +(-2.15495 + 5.44278i) q^{47} +(6.39402 + 1.19906i) q^{48} +(2.56746 + 1.86537i) q^{49} +(-6.94311 - 1.33913i) q^{50} +(0.0636951 + 0.0876687i) q^{51} +(11.7542 + 4.21108i) q^{52} +(2.02742 + 3.19470i) q^{53} +(0.127222 + 7.71539i) q^{54} +(-4.65719 - 7.49102i) q^{55} +(8.54198 + 2.90226i) q^{56} -2.09101i q^{57} +(-6.00965 - 12.2431i) q^{58} +(-5.44075 - 2.56022i) q^{59} +(-5.10326 - 5.18249i) q^{60} +(-10.4519 - 11.1301i) q^{61} +(0.321834 + 1.17283i) q^{62} +(-0.141886 + 1.12315i) q^{63} +(7.99714 + 0.213863i) q^{64} +(-8.10025 - 11.3690i) q^{65} +(0.420329 - 9.06329i) q^{66} +(-0.309930 - 4.92620i) q^{67} +(0.0943756 + 0.0940816i) q^{68} +(-3.84202 + 0.241720i) q^{69} +(-5.98745 - 8.11702i) q^{70} +(14.1537 + 5.60385i) q^{71} +(0.174910 + 0.988527i) q^{72} +(5.00080 - 2.35320i) q^{73} +(-4.72350 - 2.88950i) q^{74} +(1.37553 + 8.01466i) q^{75} +(-0.406214 - 2.53909i) q^{76} +(2.35766 - 12.3593i) q^{77} +(-0.236737 - 14.3569i) q^{78} +(6.64526 - 8.03273i) q^{79} +(-7.20364 - 5.30166i) q^{80} +(7.26086 - 2.87478i) q^{81} +(-3.84435 + 16.0681i) q^{82} +(9.27720 + 11.2142i) q^{83} +(-0.342060 - 10.3693i) q^{84} +(-0.0265610 - 0.146602i) q^{85} +(0.711628 - 6.49226i) q^{86} +(-10.7368 + 11.4335i) q^{87} +(-1.25030 - 11.0871i) q^{88} +(-2.42116 - 5.14522i) q^{89} +(0.470533 - 1.01898i) q^{90} +(2.49568 - 19.7553i) q^{91} +(-4.61838 + 1.03990i) q^{92} +(1.13152 - 0.822097i) q^{93} +(8.19510 - 1.17286i) q^{94} +(-1.24811 + 2.58983i) q^{95} +(-3.01400 - 8.69242i) q^{96} +(13.3904 + 0.842450i) q^{97} +(0.489018 - 4.46137i) 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 \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(e\left(\frac{23}{50}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.701642 − 1.22788i −0.496136 − 0.868245i
\(3\) −1.03669 + 1.25314i −0.598531 + 0.723500i −0.979580 − 0.201055i \(-0.935563\pi\)
0.381049 + 0.924555i \(0.375563\pi\)
\(4\) −1.01540 + 1.72307i −0.507698 + 0.861535i
\(5\) 2.03199 − 0.933291i 0.908732 − 0.417380i
\(6\) 2.26609 + 0.393675i 0.925128 + 0.160717i
\(7\) 3.03349 + 0.985641i 1.14655 + 0.372537i 0.819843 − 0.572588i \(-0.194060\pi\)
0.326709 + 0.945125i \(0.394060\pi\)
\(8\) 2.82817 + 0.0378094i 0.999911 + 0.0133677i
\(9\) 0.0665065 + 0.348639i 0.0221688 + 0.116213i
\(10\) −2.57170 − 1.84021i −0.813243 − 0.581924i
\(11\) −0.494407 − 3.91363i −0.149069 − 1.18000i −0.872149 − 0.489240i \(-0.837274\pi\)
0.723080 − 0.690765i \(-0.242726\pi\)
\(12\) −1.10660 − 3.05872i −0.319447 − 0.882975i
\(13\) −1.16980 − 6.13231i −0.324445 − 1.70080i −0.653862 − 0.756614i \(-0.726852\pi\)
0.329418 − 0.944184i \(-0.393148\pi\)
\(14\) −0.918173 − 4.41634i −0.245392 − 1.18032i
\(15\) −0.936990 + 3.51389i −0.241930 + 0.907282i
\(16\) −1.93794 − 3.49920i −0.484485 − 0.874799i
\(17\) 0.0165701 − 0.0645365i 0.00401885 − 0.0156524i −0.966542 − 0.256508i \(-0.917428\pi\)
0.970561 + 0.240856i \(0.0774280\pi\)
\(18\) 0.381424 − 0.326282i 0.0899026 − 0.0769054i
\(19\) −0.990641 + 0.819530i −0.227269 + 0.188013i −0.744088 − 0.668081i \(-0.767116\pi\)
0.516820 + 0.856094i \(0.327116\pi\)
\(20\) −0.455146 + 4.44891i −0.101774 + 0.994808i
\(21\) −4.37992 + 2.77958i −0.955778 + 0.606555i
\(22\) −4.45859 + 3.35305i −0.950575 + 0.714872i
\(23\) 1.62032 + 1.72547i 0.337861 + 0.359786i 0.875354 − 0.483483i \(-0.160628\pi\)
−0.537493 + 0.843268i \(0.680628\pi\)
\(24\) −2.97931 + 3.50490i −0.608149 + 0.715434i
\(25\) 3.25794 − 3.79287i 0.651587 − 0.758574i
\(26\) −6.70899 + 5.73907i −1.31574 + 1.12552i
\(27\) −4.78143 − 2.62861i −0.920187 − 0.505877i
\(28\) −4.77852 + 4.22610i −0.903056 + 0.798658i
\(29\) 9.62483 + 0.605543i 1.78729 + 0.112447i 0.921600 − 0.388140i \(-0.126882\pi\)
0.865687 + 0.500587i \(0.166882\pi\)
\(30\) 4.97208 − 1.31498i 0.907773 − 0.240081i
\(31\) −0.832957 − 0.213867i −0.149603 − 0.0384116i 0.173145 − 0.984896i \(-0.444607\pi\)
−0.322749 + 0.946485i \(0.604607\pi\)
\(32\) −2.93687 + 4.83475i −0.519169 + 0.854671i
\(33\) 5.41687 + 3.43765i 0.942956 + 0.598418i
\(34\) −0.0908696 + 0.0249353i −0.0155840 + 0.00427637i
\(35\) 7.08390 − 0.828320i 1.19740 − 0.140012i
\(36\) −0.668260 − 0.239411i −0.111377 − 0.0399019i
\(37\) 3.43109 − 1.88626i 0.564067 − 0.310099i −0.174087 − 0.984730i \(-0.555697\pi\)
0.738155 + 0.674632i \(0.235697\pi\)
\(38\) 1.70136 + 0.641376i 0.275998 + 0.104045i
\(39\) 8.89736 + 4.89136i 1.42472 + 0.783245i
\(40\) 5.78210 − 2.56268i 0.914230 − 0.405195i
\(41\) −8.51617 − 7.99722i −1.33000 − 1.24896i −0.945416 − 0.325866i \(-0.894344\pi\)
−0.384586 − 0.923089i \(-0.625656\pi\)
\(42\) 6.48615 + 3.42776i 1.00083 + 0.528915i
\(43\) 3.73622 + 2.71452i 0.569768 + 0.413961i 0.835021 − 0.550218i \(-0.185455\pi\)
−0.265253 + 0.964179i \(0.585455\pi\)
\(44\) 7.24548 + 3.12199i 1.09230 + 0.470658i
\(45\) 0.460522 + 0.646360i 0.0686505 + 0.0963537i
\(46\) 0.981789 − 3.20023i 0.144757 − 0.471849i
\(47\) −2.15495 + 5.44278i −0.314332 + 0.793911i 0.683614 + 0.729844i \(0.260407\pi\)
−0.997946 + 0.0640673i \(0.979593\pi\)
\(48\) 6.39402 + 1.19906i 0.922897 + 0.173070i
\(49\) 2.56746 + 1.86537i 0.366780 + 0.266482i
\(50\) −6.94311 − 1.33913i −0.981904 − 0.189382i
\(51\) 0.0636951 + 0.0876687i 0.00891909 + 0.0122761i
\(52\) 11.7542 + 4.21108i 1.63002 + 0.583971i
\(53\) 2.02742 + 3.19470i 0.278487 + 0.438825i 0.954564 − 0.298006i \(-0.0963214\pi\)
−0.676077 + 0.736831i \(0.736321\pi\)
\(54\) 0.127222 + 7.71539i 0.0173128 + 1.04993i
\(55\) −4.65719 − 7.49102i −0.627975 − 1.01009i
\(56\) 8.54198 + 2.90226i 1.14147 + 0.387831i
\(57\) − 2.09101i − 0.276961i
\(58\) −6.00965 − 12.2431i −0.789106 − 1.60759i
\(59\) −5.44075 − 2.56022i −0.708326 − 0.333313i 0.0376452 − 0.999291i \(-0.488014\pi\)
−0.745971 + 0.665978i \(0.768014\pi\)
\(60\) −5.10326 − 5.18249i −0.658828 − 0.669057i
\(61\) −10.4519 − 11.1301i −1.33823 − 1.42507i −0.823032 − 0.567994i \(-0.807720\pi\)
−0.515195 − 0.857073i \(-0.672280\pi\)
\(62\) 0.321834 + 1.17283i 0.0408729 + 0.148950i
\(63\) −0.141886 + 1.12315i −0.0178760 + 0.141503i
\(64\) 7.99714 + 0.213863i 0.999643 + 0.0267329i
\(65\) −8.10025 − 11.3690i −1.00471 − 1.41015i
\(66\) 0.420329 − 9.06329i 0.0517389 − 1.11561i
\(67\) −0.309930 − 4.92620i −0.0378640 − 0.601831i −0.970513 − 0.241047i \(-0.922509\pi\)
0.932649 − 0.360784i \(-0.117491\pi\)
\(68\) 0.0943756 + 0.0940816i 0.0114447 + 0.0114091i
\(69\) −3.84202 + 0.241720i −0.462525 + 0.0290996i
\(70\) −5.98745 − 8.11702i −0.715637 − 0.970170i
\(71\) 14.1537 + 5.60385i 1.67974 + 0.665055i 0.997796 − 0.0663522i \(-0.0211361\pi\)
0.681941 + 0.731407i \(0.261136\pi\)
\(72\) 0.174910 + 0.988527i 0.0206133 + 0.116499i
\(73\) 5.00080 − 2.35320i 0.585300 − 0.275421i −0.110244 − 0.993904i \(-0.535163\pi\)
0.695544 + 0.718483i \(0.255163\pi\)
\(74\) −4.72350 − 2.88950i −0.549096 − 0.335897i
\(75\) 1.37553 + 8.01466i 0.158832 + 0.925453i
\(76\) −0.406214 − 2.53909i −0.0465959 − 0.291254i
\(77\) 2.35766 − 12.3593i 0.268680 − 1.40847i
\(78\) −0.236737 − 14.3569i −0.0268052 − 1.62560i
\(79\) 6.64526 − 8.03273i 0.747650 − 0.903753i −0.250273 − 0.968175i \(-0.580520\pi\)
0.997923 + 0.0644220i \(0.0205204\pi\)
\(80\) −7.20364 − 5.30166i −0.805391 − 0.592744i
\(81\) 7.26086 − 2.87478i 0.806762 − 0.319420i
\(82\) −3.84435 + 16.0681i −0.424537 + 1.77442i
\(83\) 9.27720 + 11.2142i 1.01831 + 1.23092i 0.972726 + 0.231959i \(0.0745135\pi\)
0.0455797 + 0.998961i \(0.485487\pi\)
\(84\) −0.342060 − 10.3693i −0.0373219 − 1.13138i
\(85\) −0.0265610 − 0.146602i −0.00288094 − 0.0159012i
\(86\) 0.711628 − 6.49226i 0.0767368 − 0.700079i
\(87\) −10.7368 + 11.4335i −1.15110 + 1.22580i
\(88\) −1.25030 − 11.0871i −0.133282 − 1.18189i
\(89\) −2.42116 − 5.14522i −0.256642 − 0.545392i 0.734435 − 0.678679i \(-0.237447\pi\)
−0.991078 + 0.133286i \(0.957447\pi\)
\(90\) 0.470533 − 1.01898i 0.0495986 − 0.107410i
\(91\) 2.49568 − 19.7553i 0.261618 − 2.07092i
\(92\) −4.61838 + 1.03990i −0.481499 + 0.108417i
\(93\) 1.13152 − 0.822097i 0.117333 − 0.0852475i
\(94\) 8.19510 − 1.17286i 0.845261 − 0.120971i
\(95\) −1.24811 + 2.58983i −0.128053 + 0.265711i
\(96\) −3.01400 − 8.69242i −0.307615 − 0.887166i
\(97\) 13.3904 + 0.842450i 1.35959 + 0.0855379i 0.725765 − 0.687942i \(-0.241486\pi\)
0.633820 + 0.773480i \(0.281486\pi\)
\(98\) 0.489018 − 4.46137i 0.0493983 − 0.450666i
\(99\) 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.789.50 yes 2960
8.5 even 2 inner 1000.2.bd.a.789.86 yes 2960
125.109 even 50 inner 1000.2.bd.a.109.86 yes 2960
1000.109 even 50 inner 1000.2.bd.a.109.50 ✓ 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.109.50 ✓ 2960 1000.109 even 50 inner
1000.2.bd.a.109.86 yes 2960 125.109 even 50 inner
1000.2.bd.a.789.50 yes 2960 1.1 even 1 trivial
1000.2.bd.a.789.86 yes 2960 8.5 even 2 inner