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Properties

Label 700.2.bj.a.687.9
Level $700$
Weight $2$
Character 700.687
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 687.9
Character \(\chi\) \(=\) 700.687
Dual form 700.2.bj.a.323.9

$q$-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+(-1.35803 + 0.394648i) q^{2} +(1.07413 - 2.10809i) q^{3} +(1.68851 - 1.07189i) q^{4} +(0.402665 - 2.19951i) q^{5} +(-0.626744 + 3.28676i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(-1.87003 + 2.12203i) q^{8} +(-1.52694 - 2.10166i) q^{9} +(0.321201 + 3.14592i) q^{10} +(-2.94833 + 4.05803i) q^{11} +(-0.445973 - 4.71087i) q^{12} +(0.838617 - 5.29482i) q^{13} +(0.681216 - 1.23933i) q^{14} +(-4.20426 - 3.21141i) q^{15} +(1.70210 - 3.61978i) q^{16} +(-1.55214 + 0.790854i) q^{17} +(2.90305 + 2.25151i) q^{18} +(1.27030 - 3.90957i) q^{19} +(-1.67773 - 4.14550i) q^{20} +(0.731123 + 2.25017i) q^{21} +(2.40244 - 6.67449i) q^{22} +(-1.15788 - 7.31058i) q^{23} +(2.46478 + 6.22151i) q^{24} +(-4.67572 - 1.77134i) q^{25} +(0.950721 + 7.52150i) q^{26} +(0.939898 - 0.148865i) q^{27} +(-0.436013 + 1.95189i) q^{28} +(3.15820 - 1.02616i) q^{29} +(6.97690 + 2.70200i) q^{30} +(-6.33317 - 2.05777i) q^{31} +(-0.882973 + 5.58752i) q^{32} +(5.38781 + 10.5742i) q^{33} +(1.79575 - 1.68655i) q^{34} +(1.27056 + 1.84002i) q^{35} +(-4.83100 - 1.91195i) q^{36} +(-7.32000 - 1.15937i) q^{37} +(-0.182200 + 5.81064i) q^{38} +(-10.2612 - 7.45518i) q^{39} +(3.91443 + 4.96762i) q^{40} +(-7.79590 + 5.66405i) q^{41} +(-1.88091 - 2.76726i) q^{42} +(-1.87158 - 1.87158i) q^{43} +(-0.628514 + 10.0123i) q^{44} +(-5.23747 + 2.51227i) q^{45} +(4.45755 + 9.47105i) q^{46} +(9.67510 + 4.92971i) q^{47} +(-5.80256 - 7.47629i) q^{48} -1.00000i q^{49} +(7.04884 + 0.560267i) q^{50} +4.12152i q^{51} +(-4.25946 - 9.83924i) q^{52} +(6.97482 + 3.55385i) q^{53} +(-1.21766 + 0.573092i) q^{54} +(7.73850 + 8.11892i) q^{55} +(-0.178191 - 2.82281i) q^{56} +(-6.87727 - 6.87727i) q^{57} +(-3.88397 + 2.63994i) q^{58} +(-1.83569 + 1.33371i) q^{59} +(-10.5412 - 0.915978i) q^{60} +(8.28749 + 6.02121i) q^{61} +(9.41274 + 0.295148i) q^{62} +(2.56581 + 0.406384i) q^{63} +(-1.00600 - 7.93650i) q^{64} +(-11.3084 - 3.97659i) q^{65} +(-11.4899 - 12.2338i) q^{66} +(-3.23884 - 6.35659i) q^{67} +(-1.77309 + 2.99908i) q^{68} +(-16.6551 - 5.41156i) q^{69} +(-2.45163 - 1.99738i) q^{70} +(-1.30044 + 0.422538i) q^{71} +(7.31520 + 0.689941i) q^{72} +(-0.0266366 + 0.00421882i) q^{73} +(10.3983 - 1.31435i) q^{74} +(-8.75645 + 7.95421i) q^{75} +(-2.04573 - 7.96295i) q^{76} +(-0.784675 - 4.95424i) q^{77} +(16.8772 + 6.07483i) q^{78} +(3.40794 + 10.4886i) q^{79} +(-7.27639 - 5.20136i) q^{80} +(3.10403 - 9.55323i) q^{81} +(8.35178 - 10.7686i) q^{82} +(4.74219 - 2.41627i) q^{83} +(3.64644 + 3.01574i) q^{84} +(1.11450 + 3.73240i) q^{85} +(3.28027 + 1.80305i) q^{86} +(1.22906 - 7.76000i) q^{87} +(-3.09779 - 13.8451i) q^{88} +(1.16286 - 1.60054i) q^{89} +(6.12119 - 5.47870i) q^{90} +(3.15101 + 4.33700i) q^{91} +(-9.79123 - 11.1028i) q^{92} +(-11.1406 + 11.1406i) q^{93} +(-15.0846 - 2.87645i) q^{94} +(-8.08765 - 4.36828i) q^{95} +(10.8306 + 7.86308i) q^{96} +(5.89110 - 11.5619i) q^{97} +(0.394648 + 1.35803i) q^{98} +13.0305 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 \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{9}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.35803 + 0.394648i −0.960274 + 0.279058i
\(3\) 1.07413 − 2.10809i 0.620147 − 1.21711i −0.340738 − 0.940158i \(-0.610677\pi\)
0.960885 − 0.276948i \(-0.0893231\pi\)
\(4\) 1.68851 − 1.07189i 0.844253 − 0.535945i
\(5\) 0.402665 − 2.19951i 0.180077 − 0.983652i
\(6\) −0.626744 + 3.28676i −0.255867 + 1.34181i
\(7\) −0.707107 + 0.707107i −0.267261 + 0.267261i
\(8\) −1.87003 + 2.12203i −0.661154 + 0.750250i
\(9\) −1.52694 − 2.10166i −0.508981 − 0.700552i
\(10\) 0.321201 + 3.14592i 0.101573 + 0.994828i
\(11\) −2.94833 + 4.05803i −0.888955 + 1.22354i 0.0849041 + 0.996389i \(0.472942\pi\)
−0.973859 + 0.227153i \(0.927058\pi\)
\(12\) −0.445973 − 4.71087i −0.128741 − 1.35991i
\(13\) 0.838617 − 5.29482i 0.232591 − 1.46852i −0.544304 − 0.838888i \(-0.683206\pi\)
0.776894 − 0.629631i \(-0.216794\pi\)
\(14\) 0.681216 − 1.23933i 0.182063 − 0.331226i
\(15\) −4.20426 − 3.21141i −1.08554 − 0.829182i
\(16\) 1.70210 − 3.61978i 0.425526 − 0.904946i
\(17\) −1.55214 + 0.790854i −0.376449 + 0.191810i −0.631969 − 0.774993i \(-0.717753\pi\)
0.255521 + 0.966804i \(0.417753\pi\)
\(18\) 2.90305 + 2.25151i 0.684256 + 0.530687i
\(19\) 1.27030 − 3.90957i 0.291426 − 0.896917i −0.692973 − 0.720964i \(-0.743699\pi\)
0.984399 − 0.175953i \(-0.0563006\pi\)
\(20\) −1.67773 − 4.14550i −0.375153 − 0.926963i
\(21\) 0.731123 + 2.25017i 0.159544 + 0.491027i
\(22\) 2.40244 − 6.67449i 0.512201 − 1.42301i
\(23\) −1.15788 − 7.31058i −0.241435 − 1.52436i −0.748896 − 0.662687i \(-0.769416\pi\)
0.507461 − 0.861675i \(-0.330584\pi\)
\(24\) 2.46478 + 6.22151i 0.503121 + 1.26996i
\(25\) −4.67572 − 1.77134i −0.935144 − 0.354267i
\(26\) 0.950721 + 7.52150i 0.186452 + 1.47509i
\(27\) 0.939898 − 0.148865i 0.180883 − 0.0286491i
\(28\) −0.436013 + 1.95189i −0.0823988 + 0.368873i
\(29\) 3.15820 − 1.02616i 0.586463 − 0.190553i −0.000730781 − 1.00000i \(-0.500233\pi\)
0.587194 + 0.809446i \(0.300233\pi\)
\(30\) 6.97690 + 2.70200i 1.27380 + 0.493315i
\(31\) −6.33317 − 2.05777i −1.13747 − 0.369587i −0.321060 − 0.947059i \(-0.604039\pi\)
−0.816410 + 0.577472i \(0.804039\pi\)
\(32\) −0.882973 + 5.58752i −0.156089 + 0.987743i
\(33\) 5.38781 + 10.5742i 0.937898 + 1.84073i
\(34\) 1.79575 − 1.68655i 0.307968 − 0.289242i
\(35\) 1.27056 + 1.84002i 0.214764 + 0.311020i
\(36\) −4.83100 − 1.91195i −0.805166 − 0.318658i
\(37\) −7.32000 − 1.15937i −1.20340 − 0.190600i −0.477645 − 0.878553i \(-0.658509\pi\)
−0.725755 + 0.687953i \(0.758509\pi\)
\(38\) −0.182200 + 5.81064i −0.0295567 + 0.942611i
\(39\) −10.2612 − 7.45518i −1.64310 − 1.19378i
\(40\) 3.91443 + 4.96762i 0.618926 + 0.785449i
\(41\) −7.79590 + 5.66405i −1.21752 + 0.884577i −0.995891 − 0.0905556i \(-0.971136\pi\)
−0.221624 + 0.975132i \(0.571136\pi\)
\(42\) −1.88091 − 2.76726i −0.290231 − 0.426998i
\(43\) −1.87158 − 1.87158i −0.285413 − 0.285413i 0.549850 − 0.835263i \(-0.314685\pi\)
−0.835263 + 0.549850i \(0.814685\pi\)
\(44\) −0.628514 + 10.0123i −0.0947520 + 1.50941i
\(45\) −5.23747 + 2.51227i −0.780756 + 0.374507i
\(46\) 4.45755 + 9.47105i 0.657230 + 1.39643i
\(47\) 9.67510 + 4.92971i 1.41126 + 0.719072i 0.982832 − 0.184505i \(-0.0590682\pi\)
0.428426 + 0.903577i \(0.359068\pi\)
\(48\) −5.80256 − 7.47629i −0.837527 − 1.07911i
\(49\) − 1.00000i − 0.142857i
\(50\) 7.04884 + 0.560267i 0.996856 + 0.0792337i
\(51\) 4.12152i 0.577129i
\(52\) −4.25946 − 9.83924i −0.590680 − 1.36446i
\(53\) 6.97482 + 3.55385i 0.958065 + 0.488158i 0.861828 − 0.507200i \(-0.169320\pi\)
0.0962363 + 0.995359i \(0.469320\pi\)
\(54\) −1.21766 + 0.573092i −0.165703 + 0.0779880i
\(55\) 7.73850 + 8.11892i 1.04346 + 1.09476i
\(56\) −0.178191 − 2.82281i −0.0238118 − 0.377214i
\(57\) −6.87727 − 6.87727i −0.910916 − 0.910916i
\(58\) −3.88397 + 2.63994i −0.509990 + 0.346641i
\(59\) −1.83569 + 1.33371i −0.238986 + 0.173634i −0.700832 − 0.713327i \(-0.747188\pi\)
0.461845 + 0.886961i \(0.347188\pi\)
\(60\) −10.5412 − 0.915978i −1.36086 − 0.118252i
\(61\) 8.28749 + 6.02121i 1.06110 + 0.770937i 0.974292 − 0.225288i \(-0.0723321\pi\)
0.0868117 + 0.996225i \(0.472332\pi\)
\(62\) 9.41274 + 0.295148i 1.19542 + 0.0374839i
\(63\) 2.56581 + 0.406384i 0.323261 + 0.0511996i
\(64\) −1.00600 − 7.93650i −0.125750 − 0.992062i
\(65\) −11.3084 − 3.97659i −1.40263 − 0.493235i
\(66\) −11.4899 − 12.2338i −1.41431 − 1.50588i
\(67\) −3.23884 − 6.35659i −0.395688 − 0.776581i 0.604106 − 0.796904i \(-0.293531\pi\)
−0.999793 + 0.0203231i \(0.993531\pi\)
\(68\) −1.77309 + 2.99908i −0.215018 + 0.363692i
\(69\) −16.6551 − 5.41156i −2.00504 − 0.651476i
\(70\) −2.45163 − 1.99738i −0.293025 − 0.238733i
\(71\) −1.30044 + 0.422538i −0.154334 + 0.0501460i −0.385165 − 0.922848i \(-0.625855\pi\)
0.230831 + 0.972994i \(0.425855\pi\)
\(72\) 7.31520 + 0.689941i 0.862104 + 0.0813103i
\(73\) −0.0266366 + 0.00421882i −0.00311758 + 0.000493776i −0.157993 − 0.987440i \(-0.550502\pi\)
0.154875 + 0.987934i \(0.450502\pi\)
\(74\) 10.3983 − 1.31435i 1.20878 − 0.152791i
\(75\) −8.75645 + 7.95421i −1.01111 + 0.918473i
\(76\) −2.04573 − 7.96295i −0.234661 − 0.913413i
\(77\) −0.784675 − 4.95424i −0.0894220 − 0.564589i
\(78\) 16.8772 + 6.07483i 1.91097 + 0.687839i
\(79\) 3.40794 + 10.4886i 0.383424 + 1.18006i 0.937617 + 0.347669i \(0.113027\pi\)
−0.554194 + 0.832388i \(0.686973\pi\)
\(80\) −7.27639 − 5.20136i −0.813525 − 0.581530i
\(81\) 3.10403 − 9.55323i 0.344892 − 1.06147i
\(82\) 8.35178 − 10.7686i 0.922300 − 1.18919i
\(83\) 4.74219 − 2.41627i 0.520523 − 0.265220i −0.173930 − 0.984758i \(-0.555647\pi\)
0.694453 + 0.719538i \(0.255647\pi\)
\(84\) 3.64644 + 3.01574i 0.397859 + 0.329044i
\(85\) 1.11450 + 3.73240i 0.120885 + 0.404835i
\(86\) 3.28027 + 1.80305i 0.353721 + 0.194428i
\(87\) 1.22906 − 7.76000i 0.131769 − 0.831959i
\(88\) −3.09779 − 13.8451i −0.330225 − 1.47589i
\(89\) 1.16286 − 1.60054i 0.123263 − 0.169657i −0.742926 − 0.669374i \(-0.766563\pi\)
0.866189 + 0.499717i \(0.166563\pi\)
\(90\) 6.12119 − 5.47870i 0.645230 − 0.577506i
\(91\) 3.15101 + 4.33700i 0.330316 + 0.454641i
\(92\) −9.79123 − 11.1028i −1.02081 − 1.15755i
\(93\) −11.1406 + 11.1406i −1.15522 + 1.15522i
\(94\) −15.0846 − 2.87645i −1.55586 − 0.296683i
\(95\) −8.08765 − 4.36828i −0.829775 − 0.448176i
\(96\) 10.8306 + 7.86308i 1.10539 + 0.802522i
\(97\) 5.89110 − 11.5619i 0.598151 − 1.17394i −0.371268 − 0.928526i \(-0.621077\pi\)
0.969419 − 0.245411i \(-0.0789230\pi\)
\(98\) 0.394648 + 1.35803i 0.0398655 + 0.137182i
\(99\) 13.0305 1.30962
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.687.9 yes 720
4.3 odd 2 inner 700.2.bj.a.687.89 yes 720
25.23 odd 20 inner 700.2.bj.a.323.89 yes 720
100.23 even 20 inner 700.2.bj.a.323.9 ✓ 720
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.bj.a.323.9 ✓ 720 100.23 even 20 inner
700.2.bj.a.323.89 yes 720 25.23 odd 20 inner
700.2.bj.a.687.9 yes 720 1.1 even 1 trivial
700.2.bj.a.687.89 yes 720 4.3 odd 2 inner