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Properties

Label 1000.2.bd.a.789.95
Level $1000$
Weight $2$
Character 1000.789
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 789.95
Character \(\chi\) \(=\) 1000.789
Dual form 1000.2.bd.a.109.95

$q$-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+(0.594198 - 1.28333i) q^{2} +(-0.859509 + 1.03897i) q^{3} +(-1.29386 - 1.52510i) q^{4} +(-0.254836 - 2.22150i) q^{5} +(0.822618 + 1.72038i) q^{6} +(4.65603 + 1.51284i) q^{7} +(-2.72601 + 0.754231i) q^{8} +(0.221445 + 1.16086i) q^{9} +(-3.00233 - 0.992972i) q^{10} +(0.350697 + 2.77605i) q^{11} +(2.69661 - 0.0334388i) q^{12} +(0.189832 + 0.995136i) q^{13} +(4.70807 - 5.07629i) q^{14} +(2.52710 + 1.64463i) q^{15} +(-0.651864 + 3.94653i) q^{16} +(-1.66441 + 6.48244i) q^{17} +(1.62134 + 0.405592i) q^{18} +(1.29260 - 1.06933i) q^{19} +(-3.05829 + 3.26296i) q^{20} +(-5.57369 + 3.53717i) q^{21} +(3.77097 + 1.19947i) q^{22} +(0.102920 + 0.109598i) q^{23} +(1.55941 - 3.48051i) q^{24} +(-4.87012 + 1.13224i) q^{25} +(1.38988 + 0.347691i) q^{26} +(-4.94129 - 2.71650i) q^{27} +(-3.71702 - 9.05831i) q^{28} +(2.65084 + 0.166777i) q^{29} +(3.61220 - 2.26586i) q^{30} +(4.75898 + 1.22190i) q^{31} +(4.67735 + 3.18157i) q^{32} +(-3.18566 - 2.02168i) q^{33} +(7.33010 + 5.98783i) q^{34} +(2.17424 - 10.7289i) q^{35} +(1.48390 - 1.83971i) q^{36} +(4.28495 - 2.35567i) q^{37} +(-0.604242 - 2.29422i) q^{38} +(-1.19708 - 0.658099i) q^{39} +(2.37021 + 5.86363i) q^{40} +(2.05023 + 1.92529i) q^{41} +(1.22747 + 9.25465i) q^{42} +(-0.729437 - 0.529967i) q^{43} +(3.78001 - 4.12667i) q^{44} +(2.52241 - 0.787769i) q^{45} +(0.201805 - 0.0669565i) q^{46} +(2.32115 - 5.86254i) q^{47} +(-3.54003 - 4.06934i) q^{48} +(13.7268 + 9.97314i) q^{49} +(-1.44078 + 6.92273i) q^{50} +(-5.30447 - 7.30098i) q^{51} +(1.27207 - 1.57708i) q^{52} +(4.68128 + 7.37652i) q^{53} +(-6.42226 + 4.72716i) q^{54} +(6.07763 - 1.48651i) q^{55} +(-13.8334 - 0.612284i) q^{56} +2.26207i q^{57} +(1.78916 - 3.30280i) q^{58} +(3.28620 + 1.54637i) q^{59} +(-0.761480 - 5.98200i) q^{60} +(-8.01345 - 8.53346i) q^{61} +(4.39587 - 5.38128i) q^{62} +(-0.725130 + 5.74000i) q^{63} +(6.86227 - 4.11209i) q^{64} +(2.16232 - 0.675310i) q^{65} +(-4.48739 + 2.88696i) q^{66} +(0.269023 + 4.27600i) q^{67} +(12.0399 - 5.84896i) q^{68} +(-0.202329 + 0.0127295i) q^{69} +(-12.4768 - 9.16535i) q^{70} +(-8.35992 - 3.30993i) q^{71} +(-1.47922 - 2.99749i) q^{72} +(-0.209163 + 0.0984244i) q^{73} +(-0.476989 - 6.89873i) q^{74} +(3.00955 - 6.03307i) q^{75} +(-3.30328 - 0.587782i) q^{76} +(-2.56686 + 13.4559i) q^{77} +(-1.55586 + 1.14520i) q^{78} +(4.41669 - 5.33886i) q^{79} +(8.93332 + 0.442398i) q^{80} +(3.77304 - 1.49385i) q^{81} +(3.68902 - 1.48711i) q^{82} +(0.0671268 + 0.0811423i) q^{83} +(12.6061 + 3.92384i) q^{84} +(14.8249 + 2.04552i) q^{85} +(-1.11355 + 0.621201i) q^{86} +(-2.45170 + 2.61080i) q^{87} +(-3.04979 - 7.30305i) q^{88} +(-5.81576 - 12.3591i) q^{89} +(0.487845 - 3.70517i) q^{90} +(-0.621613 + 4.92057i) q^{91} +(0.0339850 - 0.298767i) q^{92} +(-5.35990 + 3.89419i) q^{93} +(-6.14434 - 6.46230i) q^{94} +(-2.70492 - 2.59901i) q^{95} +(-7.32578 + 2.12502i) q^{96} +(11.8143 + 0.743293i) q^{97} +(20.9553 - 11.6900i) q^{98} +(-3.14494 + 1.02185i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.594198 − 1.28333i 0.420161 − 0.907449i
\(3\) −0.859509 + 1.03897i −0.496238 + 0.599849i −0.957863 − 0.287227i \(-0.907267\pi\)
0.461625 + 0.887075i \(0.347267\pi\)
\(4\) −1.29386 − 1.52510i −0.646929 − 0.762550i
\(5\) −0.254836 − 2.22150i −0.113966 − 0.993485i
\(6\) 0.822618 + 1.72038i 0.335832 + 0.702344i
\(7\) 4.65603 + 1.51284i 1.75981 + 0.571799i 0.997181 − 0.0750320i \(-0.0239059\pi\)
0.762634 + 0.646830i \(0.223906\pi\)
\(8\) −2.72601 + 0.754231i −0.963790 + 0.266661i
\(9\) 0.221445 + 1.16086i 0.0738151 + 0.386952i
\(10\) −3.00233 − 0.992972i −0.949421 − 0.314005i
\(11\) 0.350697 + 2.77605i 0.105739 + 0.837012i 0.952394 + 0.304869i \(0.0986128\pi\)
−0.846655 + 0.532142i \(0.821387\pi\)
\(12\) 2.69661 − 0.0334388i 0.778445 − 0.00965296i
\(13\) 0.189832 + 0.995136i 0.0526500 + 0.276001i 0.998870 − 0.0475223i \(-0.0151325\pi\)
−0.946220 + 0.323524i \(0.895133\pi\)
\(14\) 4.70807 − 5.07629i 1.25828 − 1.35670i
\(15\) 2.52710 + 1.64463i 0.652495 + 0.424642i
\(16\) −0.651864 + 3.94653i −0.162966 + 0.986632i
\(17\) −1.66441 + 6.48244i −0.403678 + 1.57222i 0.364008 + 0.931396i \(0.381408\pi\)
−0.767686 + 0.640826i \(0.778592\pi\)
\(18\) 1.62134 + 0.405592i 0.382154 + 0.0955989i
\(19\) 1.29260 − 1.06933i 0.296543 − 0.245322i −0.477239 − 0.878773i \(-0.658363\pi\)
0.773782 + 0.633452i \(0.218363\pi\)
\(20\) −3.05829 + 3.26296i −0.683854 + 0.729619i
\(21\) −5.57369 + 3.53717i −1.21628 + 0.771874i
\(22\) 3.77097 + 1.19947i 0.803973 + 0.255727i
\(23\) 0.102920 + 0.109598i 0.0214602 + 0.0228528i 0.739609 − 0.673037i \(-0.235010\pi\)
−0.718149 + 0.695889i \(0.755010\pi\)
\(24\) 1.55941 − 3.48051i 0.318313 − 0.710456i
\(25\) −4.87012 + 1.13224i −0.974023 + 0.226448i
\(26\) 1.38988 + 0.347691i 0.272579 + 0.0681878i
\(27\) −4.94129 − 2.71650i −0.950952 − 0.522790i
\(28\) −3.71702 − 9.05831i −0.702450 − 1.71186i
\(29\) 2.65084 + 0.166777i 0.492249 + 0.0309697i 0.306976 − 0.951717i \(-0.400683\pi\)
0.185273 + 0.982687i \(0.440683\pi\)
\(30\) 3.61220 − 2.26586i 0.659494 − 0.413688i
\(31\) 4.75898 + 1.22190i 0.854738 + 0.219459i 0.650542 − 0.759471i \(-0.274542\pi\)
0.204196 + 0.978930i \(0.434542\pi\)
\(32\) 4.67735 + 3.18157i 0.826846 + 0.562428i
\(33\) −3.18566 − 2.02168i −0.554552 − 0.351929i
\(34\) 7.33010 + 5.98783i 1.25710 + 1.02690i
\(35\) 2.17424 − 10.7289i 0.367513 − 1.81351i
\(36\) 1.48390 − 1.83971i 0.247317 − 0.306618i
\(37\) 4.28495 − 2.35567i 0.704442 − 0.387270i −0.0889052 − 0.996040i \(-0.528337\pi\)
0.793347 + 0.608770i \(0.208337\pi\)
\(38\) −0.604242 − 2.29422i −0.0980211 − 0.372172i
\(39\) −1.19708 − 0.658099i −0.191686 − 0.105380i
\(40\) 2.37021 + 5.86363i 0.374763 + 0.927121i
\(41\) 2.05023 + 1.92529i 0.320192 + 0.300680i 0.827930 − 0.560831i \(-0.189518\pi\)
−0.507738 + 0.861511i \(0.669518\pi\)
\(42\) 1.22747 + 9.25465i 0.189403 + 1.42802i
\(43\) −0.729437 − 0.529967i −0.111238 − 0.0808192i 0.530776 − 0.847512i \(-0.321901\pi\)
−0.642014 + 0.766693i \(0.721901\pi\)
\(44\) 3.78001 − 4.12667i 0.569858 − 0.622118i
\(45\) 2.52241 − 0.787769i 0.376019 − 0.117434i
\(46\) 0.201805 − 0.0669565i 0.0297545 − 0.00987220i
\(47\) 2.32115 − 5.86254i 0.338574 − 0.855140i −0.656345 − 0.754461i \(-0.727898\pi\)
0.994919 − 0.100679i \(-0.0321015\pi\)
\(48\) −3.54003 − 4.06934i −0.510960 − 0.587359i
\(49\) 13.7268 + 9.97314i 1.96098 + 1.42473i
\(50\) −1.44078 + 6.92273i −0.203757 + 0.979021i
\(51\) −5.30447 − 7.30098i −0.742775 − 1.02234i
\(52\) 1.27207 − 1.57708i 0.176404 − 0.218701i
\(53\) 4.68128 + 7.37652i 0.643024 + 1.01324i 0.996956 + 0.0779723i \(0.0248446\pi\)
−0.353932 + 0.935271i \(0.615155\pi\)
\(54\) −6.42226 + 4.72716i −0.873959 + 0.643284i
\(55\) 6.07763 − 1.48651i 0.819507 − 0.200441i
\(56\) −13.8334 − 0.612284i −1.84857 − 0.0818199i
\(57\) 2.26207i 0.299619i
\(58\) 1.78916 − 3.30280i 0.234928 − 0.433679i
\(59\) 3.28620 + 1.54637i 0.427827 + 0.201320i 0.627623 − 0.778518i \(-0.284028\pi\)
−0.199796 + 0.979837i \(0.564028\pi\)
\(60\) −0.761480 − 5.98200i −0.0983066 − 0.772273i
\(61\) −8.01345 − 8.53346i −1.02602 − 1.09260i −0.995661 − 0.0930586i \(-0.970336\pi\)
−0.0303569 − 0.999539i \(-0.509664\pi\)
\(62\) 4.39587 − 5.38128i 0.558276 − 0.683423i
\(63\) −0.725130 + 5.74000i −0.0913578 + 0.723171i
\(64\) 6.86227 − 4.11209i 0.857784 − 0.514011i
\(65\) 2.16232 − 0.675310i 0.268203 − 0.0837619i
\(66\) −4.48739 + 2.88696i −0.552359 + 0.355361i
\(67\) 0.269023 + 4.27600i 0.0328664 + 0.522396i 0.979674 + 0.200596i \(0.0642879\pi\)
−0.946808 + 0.321800i \(0.895712\pi\)
\(68\) 12.0399 − 5.84896i 1.46005 − 0.709291i
\(69\) −0.202329 + 0.0127295i −0.0243576 + 0.00153245i
\(70\) −12.4768 − 9.16535i −1.49126 − 1.09547i
\(71\) −8.35992 − 3.30993i −0.992140 − 0.392816i −0.184719 − 0.982791i \(-0.559137\pi\)
−0.807421 + 0.589976i \(0.799137\pi\)
\(72\) −1.47922 − 2.99749i −0.174327 − 0.353257i
\(73\) −0.209163 + 0.0984244i −0.0244806 + 0.0115197i −0.437981 − 0.898984i \(-0.644306\pi\)
0.413500 + 0.910504i \(0.364306\pi\)
\(74\) −0.476989 − 6.89873i −0.0554488 − 0.801961i
\(75\) 3.00955 − 6.03307i 0.347513 − 0.696638i
\(76\) −3.30328 − 0.587782i −0.378912 − 0.0674232i
\(77\) −2.56686 + 13.4559i −0.292521 + 1.53345i
\(78\) −1.55586 + 1.14520i −0.176166 + 0.129669i
\(79\) 4.41669 − 5.33886i 0.496916 − 0.600668i −0.461112 − 0.887342i \(-0.652549\pi\)
0.958028 + 0.286674i \(0.0925495\pi\)
\(80\) 8.93332 + 0.442398i 0.998776 + 0.0494615i
\(81\) 3.77304 − 1.49385i 0.419227 − 0.165984i
\(82\) 3.68902 − 1.48711i 0.407384 − 0.164224i
\(83\) 0.0671268 + 0.0811423i 0.00736812 + 0.00890653i 0.774184 − 0.632960i \(-0.218160\pi\)
−0.766816 + 0.641867i \(0.778160\pi\)
\(84\) 12.6061 + 3.92384i 1.37544 + 0.428126i
\(85\) 14.8249 + 2.04552i 1.60798 + 0.221868i
\(86\) −1.11355 + 0.621201i −0.120077 + 0.0669858i
\(87\) −2.45170 + 2.61080i −0.262850 + 0.279907i
\(88\) −3.04979 − 7.30305i −0.325109 − 0.778507i
\(89\) −5.81576 − 12.3591i −0.616469 − 1.31006i −0.932450 − 0.361298i \(-0.882334\pi\)
0.315981 − 0.948765i \(-0.397666\pi\)
\(90\) 0.487845 − 3.70517i 0.0514234 − 0.390559i
\(91\) −0.621613 + 4.92057i −0.0651627 + 0.515816i
\(92\) 0.0339850 − 0.298767i 0.00354319 − 0.0311486i
\(93\) −5.35990 + 3.89419i −0.555796 + 0.403809i
\(94\) −6.14434 − 6.46230i −0.633740 − 0.666535i
\(95\) −2.70492 − 2.59901i −0.277519 − 0.266652i
\(96\) −7.32578 + 2.12502i −0.747684 + 0.216884i
\(97\) 11.8143 + 0.743293i 1.19956 + 0.0754700i 0.649968 − 0.759962i \(-0.274782\pi\)
0.549594 + 0.835432i \(0.314782\pi\)
\(98\) 20.9553 − 11.6900i 2.11680 − 1.18087i
\(99\) −3.14494 + 1.02185i −0.316078 + 0.102700i
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.789.95 yes 2960
8.5 even 2 inner 1000.2.bd.a.789.41 yes 2960
125.109 even 50 inner 1000.2.bd.a.109.41 ✓ 2960
1000.109 even 50 inner 1000.2.bd.a.109.95 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.109.41 ✓ 2960 125.109 even 50 inner
1000.2.bd.a.109.95 yes 2960 1000.109 even 50 inner
1000.2.bd.a.789.41 yes 2960 8.5 even 2 inner
1000.2.bd.a.789.95 yes 2960 1.1 even 1 trivial