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Properties

Label 700.2.bj.a.323.89
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 323.89
Character \(\chi\) \(=\) 700.323
Dual form 700.2.bj.a.687.89

$q$-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+(1.41352 + 0.0443226i) q^{2} +(-1.07413 - 2.10809i) q^{3} +(1.99607 + 0.125302i) q^{4} +(0.402665 + 2.19951i) q^{5} +(-1.42486 - 3.02743i) q^{6} +(0.707107 + 0.707107i) q^{7} +(2.81593 + 0.265588i) q^{8} +(-1.52694 + 2.10166i) q^{9} +(0.471687 + 3.12690i) q^{10} +(2.94833 + 4.05803i) q^{11} +(-1.87988 - 4.34249i) q^{12} +(0.838617 + 5.29482i) q^{13} +(0.968168 + 1.03085i) q^{14} +(4.20426 - 3.21141i) q^{15} +(3.96860 + 0.500222i) q^{16} +(-1.55214 - 0.790854i) q^{17} +(-2.25151 + 2.90305i) q^{18} +(-1.27030 - 3.90957i) q^{19} +(0.528145 + 4.44084i) q^{20} +(0.731123 - 2.25017i) q^{21} +(3.98766 + 5.86678i) 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} +(1.32283 + 1.50004i) q^{28} +(3.15820 + 1.02616i) q^{29} +(6.08514 - 4.35304i) q^{30} +(6.33317 - 2.05777i) q^{31} +(5.58752 + 0.882973i) q^{32} +(5.38781 - 10.5742i) q^{33} +(-2.15892 - 1.18668i) q^{34} +(-1.27056 + 1.84002i) q^{35} +(-3.31123 + 4.00373i) q^{36} +(-7.32000 + 1.15937i) q^{37} +(-1.62230 - 5.58255i) q^{38} +(10.2612 - 7.45518i) q^{39} +(0.549714 + 6.30062i) q^{40} +(-7.79590 - 5.66405i) q^{41} +(1.13319 - 3.14825i) q^{42} +(1.87158 - 1.87158i) q^{43} +(5.37660 + 8.46954i) q^{44} +(-5.23747 - 2.51227i) q^{45} +(1.96071 - 10.2823i) q^{46} +(-9.67510 + 4.92971i) q^{47} +(-3.20826 - 8.90347i) q^{48} +1.00000i q^{49} +(-6.68773 + 2.29658i) q^{50} +4.12152i q^{51} +(1.01049 + 10.6739i) q^{52} +(6.97482 - 3.55385i) q^{53} +(-1.32196 - 0.252082i) q^{54} +(-7.73850 + 8.11892i) q^{55} +(1.80336 + 2.17896i) q^{56} +(-6.87727 + 6.87727i) q^{57} +(4.41869 + 1.59048i) q^{58} +(1.83569 + 1.33371i) q^{59} +(8.79440 - 5.88340i) q^{60} +(8.28749 - 6.02121i) q^{61} +(9.04326 - 2.62800i) q^{62} +(-2.56581 + 0.406384i) q^{63} +(7.85893 + 1.49575i) q^{64} +(-11.3084 + 3.97659i) q^{65} +(8.08445 - 14.7080i) q^{66} +(3.23884 - 6.35659i) q^{67} +(-2.99908 - 1.77309i) q^{68} +(-16.6551 + 5.41156i) q^{69} +(-1.87752 + 2.54459i) q^{70} +(1.30044 + 0.422538i) q^{71} +(-4.85794 + 5.51258i) 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} +(14.8348 - 10.0832i) q^{78} +(-3.40794 + 10.4886i) q^{79} +(0.497771 + 8.93041i) q^{80} +(3.10403 + 9.55323i) q^{81} +(-10.7686 - 8.35178i) q^{82} +(-4.74219 - 2.41627i) q^{83} +(1.74132 - 4.39988i) q^{84} +(1.11450 - 3.73240i) q^{85} +(2.72846 - 2.56255i) q^{86} +(-1.22906 - 7.76000i) q^{87} +(7.22453 + 12.2102i) q^{88} +(1.16286 + 1.60054i) q^{89} +(-7.29191 - 3.78328i) q^{90} +(-3.15101 + 4.33700i) q^{91} +(3.22724 - 14.4474i) q^{92} +(-11.1406 - 11.1406i) q^{93} +(-13.8944 + 6.53941i) q^{94} +(8.08765 - 4.36828i) q^{95} +(-4.14031 - 12.7274i) q^{96} +(5.89110 + 11.5619i) q^{97} +(-0.0443226 + 1.41352i) 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 \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41352 + 0.0443226i 0.999509 + 0.0313408i
\(3\) −1.07413 − 2.10809i −0.620147 − 1.21711i −0.960885 − 0.276948i \(-0.910677\pi\)
0.340738 − 0.940158i \(-0.389323\pi\)
\(4\) 1.99607 + 0.125302i 0.998036 + 0.0626509i
\(5\) 0.402665 + 2.19951i 0.180077 + 0.983652i
\(6\) −1.42486 − 3.02743i −0.581697 − 1.23594i
\(7\) 0.707107 + 0.707107i 0.267261 + 0.267261i
\(8\) 2.81593 + 0.265588i 0.995582 + 0.0938994i
\(9\) −1.52694 + 2.10166i −0.508981 + 0.700552i
\(10\) 0.471687 + 3.12690i 0.149160 + 0.988813i
\(11\) 2.94833 + 4.05803i 0.888955 + 1.22354i 0.973859 + 0.227153i \(0.0729416\pi\)
−0.0849041 + 0.996389i \(0.527058\pi\)
\(12\) −1.87988 − 4.34249i −0.542676 − 1.25357i
\(13\) 0.838617 + 5.29482i 0.232591 + 1.46852i 0.776894 + 0.629631i \(0.216794\pi\)
−0.544304 + 0.838888i \(0.683206\pi\)
\(14\) 0.968168 + 1.03085i 0.258754 + 0.275506i
\(15\) 4.20426 − 3.21141i 1.08554 − 0.829182i
\(16\) 3.96860 + 0.500222i 0.992150 + 0.125056i
\(17\) −1.55214 − 0.790854i −0.376449 − 0.191810i 0.255521 − 0.966804i \(-0.417753\pi\)
−0.631969 + 0.774993i \(0.717753\pi\)
\(18\) −2.25151 + 2.90305i −0.530687 + 0.684256i
\(19\) −1.27030 − 3.90957i −0.291426 − 0.896917i −0.984399 − 0.175953i \(-0.943699\pi\)
0.692973 − 0.720964i \(-0.256301\pi\)
\(20\) 0.528145 + 4.44084i 0.118097 + 0.993002i
\(21\) 0.731123 − 2.25017i 0.159544 − 0.491027i
\(22\) 3.98766 + 5.86678i 0.850172 + 1.25080i
\(23\) 1.15788 − 7.31058i 0.241435 − 1.52436i −0.507461 − 0.861675i \(-0.669416\pi\)
0.748896 − 0.662687i \(-0.230584\pi\)
\(24\) −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
\(28\) 1.32283 + 1.50004i 0.249992 + 0.283480i
\(29\) 3.15820 + 1.02616i 0.586463 + 0.190553i 0.587194 − 0.809446i \(-0.300233\pi\)
−0.000730781 1.00000i \(0.500233\pi\)
\(30\) 6.08514 − 4.35304i 1.11099 − 0.794753i
\(31\) 6.33317 − 2.05777i 1.13747 − 0.369587i 0.321060 − 0.947059i \(-0.395961\pi\)
0.816410 + 0.577472i \(0.195961\pi\)
\(32\) 5.58752 + 0.882973i 0.987743 + 0.156089i
\(33\) 5.38781 − 10.5742i 0.937898 − 1.84073i
\(34\) −2.15892 − 1.18668i −0.370252 − 0.203514i
\(35\) −1.27056 + 1.84002i −0.214764 + 0.311020i
\(36\) −3.31123 + 4.00373i −0.551871 + 0.667288i
\(37\) −7.32000 + 1.15937i −1.20340 + 0.190600i −0.725755 − 0.687953i \(-0.758509\pi\)
−0.477645 + 0.878553i \(0.658509\pi\)
\(38\) −1.62230 − 5.58255i −0.263173 − 0.905610i
\(39\) 10.2612 − 7.45518i 1.64310 − 1.19378i
\(40\) 0.549714 + 6.30062i 0.0869174 + 0.996216i
\(41\) −7.79590 − 5.66405i −1.21752 − 0.884577i −0.221624 − 0.975132i \(-0.571136\pi\)
−0.995891 + 0.0905556i \(0.971136\pi\)
\(42\) 1.13319 − 3.14825i 0.174855 − 0.485785i
\(43\) 1.87158 − 1.87158i 0.285413 − 0.285413i −0.549850 − 0.835263i \(-0.685315\pi\)
0.835263 + 0.549850i \(0.185315\pi\)
\(44\) 5.37660 + 8.46954i 0.810553 + 1.27683i
\(45\) −5.23747 − 2.51227i −0.780756 − 0.374507i
\(46\) 1.96071 − 10.2823i 0.289091 − 1.51605i
\(47\) −9.67510 + 4.92971i −1.41126 + 0.719072i −0.982832 − 0.184505i \(-0.940932\pi\)
−0.428426 + 0.903577i \(0.640932\pi\)
\(48\) −3.20826 − 8.90347i −0.463072 − 1.28510i
\(49\) 1.00000i 0.142857i
\(50\) −6.68773 + 2.29658i −0.945788 + 0.324785i
\(51\) 4.12152i 0.577129i
\(52\) 1.01049 + 10.6739i 0.140130 + 1.48021i
\(53\) 6.97482 − 3.55385i 0.958065 − 0.488158i 0.0962363 − 0.995359i \(-0.469320\pi\)
0.861828 + 0.507200i \(0.169320\pi\)
\(54\) −1.32196 − 0.252082i −0.179897 − 0.0343041i
\(55\) −7.73850 + 8.11892i −1.04346 + 1.09476i
\(56\) 1.80336 + 2.17896i 0.240985 + 0.291176i
\(57\) −6.87727 + 6.87727i −0.910916 + 0.910916i
\(58\) 4.41869 + 1.59048i 0.580203 + 0.208840i
\(59\) 1.83569 + 1.33371i 0.238986 + 0.173634i 0.700832 − 0.713327i \(-0.252812\pi\)
−0.461845 + 0.886961i \(0.652812\pi\)
\(60\) 8.79440 − 5.88340i 1.13535 − 0.759544i
\(61\) 8.28749 − 6.02121i 1.06110 − 0.770937i 0.0868117 − 0.996225i \(-0.472332\pi\)
0.974292 + 0.225288i \(0.0723321\pi\)
\(62\) 9.04326 − 2.62800i 1.14849 − 0.333756i
\(63\) −2.56581 + 0.406384i −0.323261 + 0.0511996i
\(64\) 7.85893 + 1.49575i 0.982366 + 0.186969i
\(65\) −11.3084 + 3.97659i −1.40263 + 0.493235i
\(66\) 8.08445 − 14.7080i 0.995127 − 1.81043i
\(67\) 3.23884 − 6.35659i 0.395688 − 0.776581i −0.604106 − 0.796904i \(-0.706469\pi\)
0.999793 + 0.0203231i \(0.00646949\pi\)
\(68\) −2.99908 − 1.77309i −0.363692 − 0.215018i
\(69\) −16.6551 + 5.41156i −2.00504 + 0.651476i
\(70\) −1.87752 + 2.54459i −0.224407 + 0.304136i
\(71\) 1.30044 + 0.422538i 0.154334 + 0.0501460i 0.385165 − 0.922848i \(-0.374145\pi\)
−0.230831 + 0.972994i \(0.574145\pi\)
\(72\) −4.85794 + 5.51258i −0.572514 + 0.649664i
\(73\) −0.0266366 − 0.00421882i −0.00311758 − 0.000493776i 0.154875 − 0.987934i \(-0.450502\pi\)
−0.157993 + 0.987440i \(0.550502\pi\)
\(74\) −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
\(78\) 14.8348 − 10.0832i 1.67971 − 1.14170i
\(79\) −3.40794 + 10.4886i −0.383424 + 1.18006i 0.554194 + 0.832388i \(0.313027\pi\)
−0.937617 + 0.347669i \(0.886973\pi\)
\(80\) 0.497771 + 8.93041i 0.0556525 + 0.998450i
\(81\) 3.10403 + 9.55323i 0.344892 + 1.06147i
\(82\) −10.7686 − 8.35178i −1.18919 − 0.922300i
\(83\) −4.74219 − 2.41627i −0.520523 − 0.265220i 0.173930 − 0.984758i \(-0.444353\pi\)
−0.694453 + 0.719538i \(0.744353\pi\)
\(84\) 1.74132 − 4.39988i 0.189994 − 0.480066i
\(85\) 1.11450 − 3.73240i 0.120885 − 0.404835i
\(86\) 2.72846 − 2.56255i 0.294217 − 0.276327i
\(87\) −1.22906 − 7.76000i −0.131769 − 0.831959i
\(88\) 7.22453 + 12.2102i 0.770138 + 1.30161i
\(89\) 1.16286 + 1.60054i 0.123263 + 0.169657i 0.866189 − 0.499717i \(-0.166563\pi\)
−0.742926 + 0.669374i \(0.766563\pi\)
\(90\) −7.29191 − 3.78328i −0.768635 − 0.398792i
\(91\) −3.15101 + 4.33700i −0.330316 + 0.454641i
\(92\) 3.22724 − 14.4474i 0.336463 − 1.50624i
\(93\) −11.1406 − 11.1406i −1.15522 − 1.15522i
\(94\) −13.8944 + 6.53941i −1.43310 + 0.674489i
\(95\) 8.08765 − 4.36828i 0.829775 − 0.448176i
\(96\) −4.14031 − 12.7274i −0.422569 − 1.29899i
\(97\) 5.89110 + 11.5619i 0.598151 + 1.17394i 0.969419 + 0.245411i \(0.0789230\pi\)
−0.371268 + 0.928526i \(0.621077\pi\)
\(98\) −0.0443226 + 1.41352i −0.00447726 + 0.142787i
\(99\) −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.323.89 yes 720
4.3 odd 2 inner 700.2.bj.a.323.9 ✓ 720
25.12 odd 20 inner 700.2.bj.a.687.9 yes 720
100.87 even 20 inner 700.2.bj.a.687.89 yes 720
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.bj.a.323.9 ✓ 720 4.3 odd 2 inner
700.2.bj.a.323.89 yes 720 1.1 even 1 trivial
700.2.bj.a.687.9 yes 720 25.12 odd 20 inner
700.2.bj.a.687.89 yes 720 100.87 even 20 inner