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Properties

Label 700.2.bj.a.323.32
Level $700$
Weight $2$
Character 700.323
Analytic conductor $5.590$
Analytic rank $0$
Dimension $720$
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] = [700,2,Mod(127,700)] 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("700.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.bj (of order \(20\), degree \(8\), 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(720\)
Relative dimension: \(90\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 323.32
Character \(\chi\) \(=\) 700.323
Dual form 700.2.bj.a.687.32

$q$-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+(-0.521152 - 1.31469i) q^{2} +(1.38963 + 2.72730i) q^{3} +(-1.45680 + 1.37030i) q^{4} +(-0.550371 - 2.16728i) q^{5} +(2.86134 - 3.24827i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(2.56073 + 1.20110i) q^{8} +(-3.74375 + 5.15283i) q^{9} +(-2.56246 + 1.85305i) q^{10} +(0.474640 + 0.653285i) q^{11} +(-5.76164 - 2.06892i) q^{12} +(-0.0801358 - 0.505957i) q^{13} +(-0.561113 + 1.29813i) q^{14} +(5.14601 - 4.51274i) q^{15} +(0.244537 - 3.99252i) q^{16} +(6.88020 + 3.50564i) q^{17} +(8.72542 + 2.23645i) q^{18} +(2.40162 + 7.39142i) q^{19} +(3.77161 + 2.40312i) q^{20} +(0.945877 - 2.91111i) q^{21} +(0.611506 - 0.964463i) q^{22} +(-0.516845 + 3.26323i) q^{23} +(0.282711 + 8.65298i) q^{24} +(-4.39418 + 2.38562i) q^{25} +(-0.623412 + 0.369034i) q^{26} +(-10.1860 - 1.61331i) q^{27} +(1.99906 + 0.0611629i) q^{28} +(3.83855 + 1.24722i) q^{29} +(-8.61469 - 4.41356i) q^{30} +(-2.71641 + 0.882616i) q^{31} +(-5.37635 + 1.75922i) q^{32} +(-1.12213 + 2.20231i) q^{33} +(1.02318 - 10.8723i) q^{34} +(-1.14333 + 1.92167i) q^{35} +(-1.60704 - 12.6367i) q^{36} +(-0.458943 + 0.0726894i) q^{37} +(8.46579 - 7.00943i) q^{38} +(1.26854 - 0.921648i) q^{39} +(1.19376 - 6.21087i) q^{40} +(-1.69993 - 1.23507i) q^{41} +(-4.32014 + 0.273600i) q^{42} +(-3.73632 + 3.73632i) q^{43} +(-1.58665 - 0.301306i) q^{44} +(13.2281 + 5.27777i) q^{45} +(4.55948 - 1.02115i) q^{46} +(-0.121322 + 0.0618164i) q^{47} +(11.2286 - 4.88119i) q^{48} +1.00000i q^{49} +(5.42637 + 4.53370i) q^{50} +23.6359i q^{51} +(0.810057 + 0.627269i) q^{52} +(-10.1551 + 5.17428i) q^{53} +(3.18748 + 14.2322i) q^{54} +(1.15462 - 1.38823i) q^{55} +(-0.961407 - 2.66002i) q^{56} +(-16.8213 + 16.8213i) q^{57} +(-0.360765 - 5.69648i) q^{58} +(-2.93833 - 2.13482i) q^{59} +(-1.31288 + 13.6258i) q^{60} +(6.80234 - 4.94219i) q^{61} +(2.57603 + 3.11125i) q^{62} +(6.29083 - 0.996369i) q^{63} +(5.11472 + 6.15139i) q^{64} +(-1.05245 + 0.452141i) q^{65} +(3.48015 + 0.327514i) q^{66} +(5.45237 - 10.7009i) q^{67} +(-14.8269 + 4.32095i) q^{68} +(-9.61804 + 3.12509i) q^{69} +(3.12224 + 0.501633i) q^{70} +(11.4121 + 3.70801i) q^{71} +(-15.7758 + 8.69841i) q^{72} +(7.94328 + 1.25809i) q^{73} +(0.334743 + 0.565484i) q^{74} +(-12.6126 - 8.66914i) q^{75} +(-13.6272 - 7.47688i) q^{76} +(0.126322 - 0.797563i) q^{77} +(-1.87278 - 1.18741i) q^{78} +(2.04132 - 6.28253i) q^{79} +(-8.78748 + 1.66739i) q^{80} +(-3.85021 - 11.8497i) q^{81} +(-0.737811 + 2.87854i) q^{82} +(8.41964 + 4.29002i) q^{83} +(2.61115 + 5.53705i) q^{84} +(3.81102 - 16.8407i) q^{85} +(6.85927 + 2.96490i) q^{86} +(1.93262 + 12.2021i) q^{87} +(0.430765 + 2.24298i) q^{88} +(-8.53762 - 11.7510i) q^{89} +(0.0447832 - 20.1413i) q^{90} +(-0.301101 + 0.414430i) q^{91} +(-3.71868 - 5.46211i) q^{92} +(-6.18197 - 6.18197i) q^{93} +(0.144496 + 0.127284i) q^{94} +(14.6975 - 9.27300i) q^{95} +(-12.2691 - 12.2183i) q^{96} +(-0.880373 - 1.72783i) q^{97} +(1.31469 - 0.521152i) q^{98} -5.14320 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 720 q + 16 q^{10} + 16 q^{12} + 4 q^{13} + 20 q^{17} - 28 q^{18} - 20 q^{20} - 4 q^{22} + 20 q^{25} + 4 q^{30} - 20 q^{37} - 64 q^{40} - 80 q^{42} - 140 q^{44} - 20 q^{45} - 236 q^{48} - 40 q^{50} - 16 q^{52}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{11}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.521152 − 1.31469i −0.368510 − 0.929624i
\(3\) 1.38963 + 2.72730i 0.802303 + 1.57461i 0.818338 + 0.574737i \(0.194896\pi\)
−0.0160352 + 0.999871i \(0.505104\pi\)
\(4\) −1.45680 + 1.37030i −0.728400 + 0.685152i
\(5\) −0.550371 − 2.16728i −0.246134 − 0.969236i
\(6\) 2.86134 − 3.24827i 1.16814 − 1.32610i
\(7\) −0.707107 − 0.707107i −0.267261 − 0.267261i
\(8\) 2.56073 + 1.20110i 0.905356 + 0.424653i
\(9\) −3.74375 + 5.15283i −1.24792 + 1.71761i
\(10\) −2.56246 + 1.85305i −0.810322 + 0.585985i
\(11\) 0.474640 + 0.653285i 0.143109 + 0.196973i 0.874555 − 0.484927i \(-0.161154\pi\)
−0.731445 + 0.681900i \(0.761154\pi\)
\(12\) −5.76164 − 2.06892i −1.66324 − 0.597246i
\(13\) −0.0801358 − 0.505957i −0.0222257 − 0.140327i 0.974080 − 0.226203i \(-0.0726312\pi\)
−0.996306 + 0.0858756i \(0.972631\pi\)
\(14\) −0.561113 + 1.29813i −0.149964 + 0.346941i
\(15\) 5.14601 − 4.51274i 1.32869 − 1.16518i
\(16\) 0.244537 − 3.99252i 0.0611343 − 0.998130i
\(17\) 6.88020 + 3.50564i 1.66869 + 0.850242i 0.993656 + 0.112458i \(0.0358725\pi\)
0.675037 + 0.737784i \(0.264128\pi\)
\(18\) 8.72542 + 2.23645i 2.05660 + 0.527136i
\(19\) 2.40162 + 7.39142i 0.550969 + 1.69571i 0.706359 + 0.707854i \(0.250337\pi\)
−0.155390 + 0.987853i \(0.549663\pi\)
\(20\) 3.77161 + 2.40312i 0.843357 + 0.537353i
\(21\) 0.945877 − 2.91111i 0.206407 − 0.635256i
\(22\) 0.611506 − 0.964463i 0.130373 − 0.205624i
\(23\) −0.516845 + 3.26323i −0.107770 + 0.680431i 0.873359 + 0.487076i \(0.161937\pi\)
−0.981129 + 0.193354i \(0.938063\pi\)
\(24\) 0.282711 + 8.65298i 0.0577081 + 1.76628i
\(25\) −4.39418 + 2.38562i −0.878836 + 0.477123i
\(26\) −0.623412 + 0.369034i −0.122261 + 0.0723736i
\(27\) −10.1860 − 1.61331i −1.96030 − 0.310481i
\(28\) 1.99906 + 0.0611629i 0.377788 + 0.0115587i
\(29\) 3.83855 + 1.24722i 0.712801 + 0.231603i 0.642899 − 0.765951i \(-0.277731\pi\)
0.0699017 + 0.997554i \(0.477731\pi\)
\(30\) −8.61469 − 4.41356i −1.57282 − 0.805802i
\(31\) −2.71641 + 0.882616i −0.487882 + 0.158523i −0.542620 − 0.839978i \(-0.682568\pi\)
0.0547380 + 0.998501i \(0.482568\pi\)
\(32\) −5.37635 + 1.75922i −0.950413 + 0.310989i
\(33\) −1.12213 + 2.20231i −0.195338 + 0.383373i
\(34\) 1.02318 − 10.8723i 0.175474 − 1.86458i
\(35\) −1.14333 + 1.92167i −0.193257 + 0.324821i
\(36\) −1.60704 − 12.6367i −0.267840 − 2.10612i
\(37\) −0.458943 + 0.0726894i −0.0754497 + 0.0119501i −0.194045 − 0.980993i \(-0.562161\pi\)
0.118595 + 0.992943i \(0.462161\pi\)
\(38\) 8.46579 − 7.00943i 1.37333 − 1.13708i
\(39\) 1.26854 − 0.921648i 0.203129 − 0.147582i
\(40\) 1.19376 − 6.21087i 0.188750 − 0.982025i
\(41\) −1.69993 − 1.23507i −0.265485 − 0.192886i 0.447077 − 0.894496i \(-0.352465\pi\)
−0.712562 + 0.701609i \(0.752465\pi\)
\(42\) −4.32014 + 0.273600i −0.666613 + 0.0422173i
\(43\) −3.73632 + 3.73632i −0.569783 + 0.569783i −0.932068 − 0.362285i \(-0.881997\pi\)
0.362285 + 0.932068i \(0.381997\pi\)
\(44\) −1.58665 − 0.301306i −0.239197 − 0.0454236i
\(45\) 13.2281 + 5.27777i 1.97192 + 0.786764i
\(46\) 4.55948 − 1.02115i 0.672259 − 0.150560i
\(47\) −0.121322 + 0.0618164i −0.0176966 + 0.00901685i −0.462816 − 0.886454i \(-0.653161\pi\)
0.445120 + 0.895471i \(0.353161\pi\)
\(48\) 11.2286 − 4.88119i 1.62071 − 0.704540i
\(49\) 1.00000i 0.142857i
\(50\) 5.42637 + 4.53370i 0.767405 + 0.641162i
\(51\) 23.6359i 3.30969i
\(52\) 0.810057 + 0.627269i 0.112335 + 0.0869865i
\(53\) −10.1551 + 5.17428i −1.39491 + 0.710742i −0.979977 − 0.199110i \(-0.936195\pi\)
−0.414933 + 0.909852i \(0.636195\pi\)
\(54\) 3.18748 + 14.2322i 0.433761 + 1.93676i
\(55\) 1.15462 − 1.38823i 0.155689 − 0.187188i
\(56\) −0.961407 − 2.66002i −0.128473 − 0.355460i
\(57\) −16.8213 + 16.8213i −2.22803 + 2.22803i
\(58\) −0.360765 − 5.69648i −0.0473707 − 0.747985i
\(59\) −2.93833 − 2.13482i −0.382537 − 0.277930i 0.379853 − 0.925047i \(-0.375974\pi\)
−0.762391 + 0.647117i \(0.775974\pi\)
\(60\) −1.31288 + 13.6258i −0.169492 + 1.75908i
\(61\) 6.80234 − 4.94219i 0.870951 − 0.632783i −0.0598912 − 0.998205i \(-0.519075\pi\)
0.930842 + 0.365422i \(0.119075\pi\)
\(62\) 2.57603 + 3.11125i 0.327156 + 0.395130i
\(63\) 6.29083 − 0.996369i 0.792570 − 0.125531i
\(64\) 5.11472 + 6.15139i 0.639340 + 0.768924i
\(65\) −1.05245 + 0.452141i −0.130540 + 0.0560812i
\(66\) 3.48015 + 0.327514i 0.428377 + 0.0403142i
\(67\) 5.45237 − 10.7009i 0.666113 − 1.30732i −0.272436 − 0.962174i \(-0.587829\pi\)
0.938549 − 0.345146i \(-0.112171\pi\)
\(68\) −14.8269 + 4.32095i −1.79802 + 0.523992i
\(69\) −9.61804 + 3.12509i −1.15788 + 0.376217i
\(70\) 3.12224 + 0.501633i 0.373179 + 0.0599566i
\(71\) 11.4121 + 3.70801i 1.35437 + 0.440060i 0.894158 − 0.447751i \(-0.147775\pi\)
0.460208 + 0.887811i \(0.347775\pi\)
\(72\) −15.7758 + 8.69841i −1.85920 + 1.02512i
\(73\) 7.94328 + 1.25809i 0.929691 + 0.147249i 0.602871 − 0.797838i \(-0.294023\pi\)
0.326819 + 0.945087i \(0.394023\pi\)
\(74\) 0.334743 + 0.565484i 0.0389131 + 0.0657361i
\(75\) −12.6126 − 8.66914i −1.45638 − 1.00103i
\(76\) −13.6272 − 7.47688i −1.56314 − 0.857657i
\(77\) 0.126322 − 0.797563i 0.0143957 − 0.0908908i
\(78\) −1.87278 − 1.18741i −0.212051 − 0.134448i
\(79\) 2.04132 − 6.28253i 0.229666 − 0.706840i −0.768118 − 0.640308i \(-0.778807\pi\)
0.997784 − 0.0665318i \(-0.0211934\pi\)
\(80\) −8.78748 + 1.66739i −0.982470 + 0.186420i
\(81\) −3.85021 − 11.8497i −0.427801 − 1.31664i
\(82\) −0.737811 + 2.87854i −0.0814776 + 0.317882i
\(83\) 8.41964 + 4.29002i 0.924176 + 0.470891i 0.850254 − 0.526373i \(-0.176448\pi\)
0.0739220 + 0.997264i \(0.476448\pi\)
\(84\) 2.61115 + 5.53705i 0.284900 + 0.604141i
\(85\) 3.81102 − 16.8407i 0.413363 − 1.82663i
\(86\) 6.85927 + 2.96490i 0.739655 + 0.319713i
\(87\) 1.93262 + 12.2021i 0.207198 + 1.30820i
\(88\) 0.430765 + 2.24298i 0.0459197 + 0.239102i
\(89\) −8.53762 − 11.7510i −0.904986 − 1.24561i −0.968850 − 0.247647i \(-0.920343\pi\)
0.0638645 − 0.997959i \(-0.479657\pi\)
\(90\) 0.0447832 − 20.1413i 0.00472056 − 2.12308i
\(91\) −0.301101 + 0.414430i −0.0315640 + 0.0434441i
\(92\) −3.71868 − 5.46211i −0.387699 − 0.569464i
\(93\) −6.18197 − 6.18197i −0.641040 − 0.641040i
\(94\) 0.144496 + 0.127284i 0.0149036 + 0.0131283i
\(95\) 14.6975 − 9.27300i 1.50793 − 0.951389i
\(96\) −12.2691 − 12.2183i −1.25221 − 1.24702i
\(97\) −0.880373 − 1.72783i −0.0893883 − 0.175434i 0.841984 − 0.539502i \(-0.181388\pi\)
−0.931372 + 0.364068i \(0.881388\pi\)
\(98\) 1.31469 − 0.521152i 0.132803 − 0.0526443i
\(99\) −5.14320 −0.516911
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 700.2.bj.a.323.32 ✓ 720
4.3 odd 2 inner 700.2.bj.a.323.66 yes 720
25.12 odd 20 inner 700.2.bj.a.687.66 yes 720
100.87 even 20 inner 700.2.bj.a.687.32 yes 720
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.bj.a.323.32 ✓ 720 1.1 even 1 trivial
700.2.bj.a.323.66 yes 720 4.3 odd 2 inner
700.2.bj.a.687.32 yes 720 100.87 even 20 inner
700.2.bj.a.687.66 yes 720 25.12 odd 20 inner