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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [800,2,Mod(63,800)] 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("800.63"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 0, 19])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.bq (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56,0,0,0,-2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(7\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 223.3
Character \(\chi\) \(=\) 800.223
Dual form 800.2.bq.b.287.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.640680 - 1.25741i) q^{3} +(-2.21664 + 0.294142i) q^{5} +(1.10768 + 1.10768i) q^{7} +(0.592758 - 0.815862i) q^{9} +(0.819489 + 1.12793i) q^{11} +(-0.463819 - 2.92844i) q^{13} +(1.79001 + 2.59876i) q^{15} +(-5.05009 - 2.57315i) q^{17} +(-1.40461 - 4.32296i) q^{19} +(0.683135 - 2.10247i) q^{21} +(-1.16984 + 7.38607i) q^{23} +(4.82696 - 1.30401i) q^{25} +(-5.58717 - 0.884921i) q^{27} +(-7.48327 - 2.43146i) q^{29} +(0.925154 - 0.300601i) q^{31} +(0.893235 - 1.75307i) q^{33} +(-2.78114 - 2.12951i) q^{35} +(-3.20334 + 0.507360i) q^{37} +(-3.38507 + 2.45940i) q^{39} +(-2.94610 - 2.14046i) q^{41} +(-6.14121 + 6.14121i) q^{43} +(-1.07395 + 1.98282i) q^{45} +(-7.68726 + 3.91685i) q^{47} -4.54608i q^{49} +7.99858i q^{51} +(6.99350 - 3.56337i) q^{53} +(-2.14828 - 2.25916i) q^{55} +(-4.53580 + 4.53580i) q^{57} +(-1.76505 - 1.28238i) q^{59} +(-8.75440 + 6.36044i) q^{61} +(1.56030 - 0.247128i) q^{63} +(1.88949 + 6.35485i) q^{65} +(3.94751 - 7.74742i) q^{67} +(10.0368 - 3.26115i) q^{69} +(9.59589 + 3.11789i) q^{71} +(-6.06170 - 0.960079i) q^{73} +(-4.73221 - 5.23399i) q^{75} +(-0.341654 + 2.15712i) q^{77} +(2.55159 - 7.85300i) q^{79} +(1.53199 + 4.71498i) q^{81} +(-7.21554 - 3.67650i) q^{83} +(11.9511 + 4.21830i) q^{85} +(1.73705 + 10.9673i) q^{87} +(2.35048 + 3.23515i) q^{89} +(2.73001 - 3.75754i) q^{91} +(-0.970704 - 0.970704i) q^{93} +(4.38508 + 9.16927i) q^{95} +(-2.03227 - 3.98855i) q^{97} +1.40599 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{5} + 4 q^{7} - 6 q^{13} + 42 q^{15} - 2 q^{17} + 18 q^{19} + 16 q^{21} + 8 q^{23} - 34 q^{25} - 42 q^{27} - 20 q^{31} - 36 q^{33} - 22 q^{35} + 20 q^{37} + 36 q^{39} + 16 q^{41} + 32 q^{43}+ \cdots - 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{11}{20}\right)\)

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 0
\(3\) −0.640680 − 1.25741i −0.369897 − 0.725963i 0.628770 − 0.777592i \(-0.283559\pi\)
−0.998666 + 0.0516283i \(0.983559\pi\)
\(4\) 0 0
\(5\) −2.21664 + 0.294142i −0.991310 + 0.131544i
\(6\) 0 0
\(7\) 1.10768 + 1.10768i 0.418664 + 0.418664i 0.884743 − 0.466079i \(-0.154334\pi\)
−0.466079 + 0.884743i \(0.654334\pi\)
\(8\) 0 0
\(9\) 0.592758 − 0.815862i 0.197586 − 0.271954i
\(10\) 0 0
\(11\) 0.819489 + 1.12793i 0.247085 + 0.340084i 0.914488 − 0.404614i \(-0.132594\pi\)
−0.667403 + 0.744697i \(0.732594\pi\)
\(12\) 0 0
\(13\) −0.463819 − 2.92844i −0.128640 − 0.812202i −0.964659 − 0.263501i \(-0.915123\pi\)
0.836019 − 0.548701i \(-0.184877\pi\)
\(14\) 0 0
\(15\) 1.79001 + 2.59876i 0.462179 + 0.670997i
\(16\) 0 0
\(17\) −5.05009 − 2.57315i −1.22483 − 0.624081i −0.282659 − 0.959220i \(-0.591217\pi\)
−0.942169 + 0.335139i \(0.891217\pi\)
\(18\) 0 0
\(19\) −1.40461 − 4.32296i −0.322241 − 0.991755i −0.972671 − 0.232189i \(-0.925411\pi\)
0.650430 − 0.759566i \(-0.274589\pi\)
\(20\) 0 0
\(21\) 0.683135 − 2.10247i 0.149072 − 0.458797i
\(22\) 0 0
\(23\) −1.16984 + 7.38607i −0.243928 + 1.54010i 0.496542 + 0.868013i \(0.334603\pi\)
−0.740470 + 0.672089i \(0.765397\pi\)
\(24\) 0 0
\(25\) 4.82696 − 1.30401i 0.965392 − 0.260802i
\(26\) 0 0
\(27\) −5.58717 − 0.884921i −1.07525 − 0.170303i
\(28\) 0 0
\(29\) −7.48327 − 2.43146i −1.38961 − 0.451511i −0.483792 − 0.875183i \(-0.660741\pi\)
−0.905816 + 0.423672i \(0.860741\pi\)
\(30\) 0 0
\(31\) 0.925154 − 0.300601i 0.166163 − 0.0539895i −0.224754 − 0.974415i \(-0.572158\pi\)
0.390917 + 0.920426i \(0.372158\pi\)
\(32\) 0 0
\(33\) 0.893235 − 1.75307i 0.155492 − 0.305171i
\(34\) 0 0
\(35\) −2.78114 − 2.12951i −0.470099 − 0.359953i
\(36\) 0 0
\(37\) −3.20334 + 0.507360i −0.526626 + 0.0834094i −0.414086 − 0.910238i \(-0.635899\pi\)
−0.112540 + 0.993647i \(0.535899\pi\)
\(38\) 0 0
\(39\) −3.38507 + 2.45940i −0.542046 + 0.393819i
\(40\) 0 0
\(41\) −2.94610 − 2.14046i −0.460103 − 0.334284i 0.333469 − 0.942761i \(-0.391781\pi\)
−0.793572 + 0.608477i \(0.791781\pi\)
\(42\) 0 0
\(43\) −6.14121 + 6.14121i −0.936526 + 0.936526i −0.998102 − 0.0615762i \(-0.980387\pi\)
0.0615762 + 0.998102i \(0.480387\pi\)
\(44\) 0 0
\(45\) −1.07395 + 1.98282i −0.160095 + 0.295582i
\(46\) 0 0
\(47\) −7.68726 + 3.91685i −1.12130 + 0.571332i −0.913500 − 0.406839i \(-0.866631\pi\)
−0.207802 + 0.978171i \(0.566631\pi\)
\(48\) 0 0
\(49\) − 4.54608i − 0.649441i
\(50\) 0 0
\(51\) 7.99858i 1.12003i
\(52\) 0 0
\(53\) 6.99350 − 3.56337i 0.960631 − 0.489466i 0.0979372 − 0.995193i \(-0.468776\pi\)
0.862694 + 0.505727i \(0.168776\pi\)
\(54\) 0 0
\(55\) −2.14828 − 2.25916i −0.289674 − 0.304626i
\(56\) 0 0
\(57\) −4.53580 + 4.53580i −0.600782 + 0.600782i
\(58\) 0 0
\(59\) −1.76505 − 1.28238i −0.229789 − 0.166952i 0.466933 − 0.884293i \(-0.345359\pi\)
−0.696722 + 0.717341i \(0.745359\pi\)
\(60\) 0 0
\(61\) −8.75440 + 6.36044i −1.12089 + 0.814371i −0.984343 − 0.176263i \(-0.943599\pi\)
−0.136543 + 0.990634i \(0.543599\pi\)
\(62\) 0 0
\(63\) 1.56030 − 0.247128i 0.196580 − 0.0311351i
\(64\) 0 0
\(65\) 1.88949 + 6.35485i 0.234363 + 0.788223i
\(66\) 0 0
\(67\) 3.94751 − 7.74742i 0.482265 − 0.946498i −0.513803 − 0.857908i \(-0.671764\pi\)
0.996068 − 0.0885899i \(-0.0282361\pi\)
\(68\) 0 0
\(69\) 10.0368 − 3.26115i 1.20829 − 0.392596i
\(70\) 0 0
\(71\) 9.59589 + 3.11789i 1.13882 + 0.370026i 0.816922 − 0.576748i \(-0.195678\pi\)
0.321900 + 0.946774i \(0.395678\pi\)
\(72\) 0 0
\(73\) −6.06170 − 0.960079i −0.709468 − 0.112369i −0.208737 − 0.977972i \(-0.566935\pi\)
−0.500731 + 0.865603i \(0.666935\pi\)
\(74\) 0 0
\(75\) −4.73221 − 5.23399i −0.546428 − 0.604370i
\(76\) 0 0
\(77\) −0.341654 + 2.15712i −0.0389351 + 0.245826i
\(78\) 0 0
\(79\) 2.55159 − 7.85300i 0.287077 − 0.883531i −0.698692 − 0.715423i \(-0.746234\pi\)
0.985768 − 0.168109i \(-0.0537659\pi\)
\(80\) 0 0
\(81\) 1.53199 + 4.71498i 0.170221 + 0.523887i
\(82\) 0 0
\(83\) −7.21554 − 3.67650i −0.792008 − 0.403548i 0.0106832 − 0.999943i \(-0.496599\pi\)
−0.802691 + 0.596395i \(0.796599\pi\)
\(84\) 0 0
\(85\) 11.9511 + 4.21830i 1.29628 + 0.457539i
\(86\) 0 0
\(87\) 1.73705 + 10.9673i 0.186231 + 1.17582i
\(88\) 0 0
\(89\) 2.35048 + 3.23515i 0.249150 + 0.342925i 0.915213 − 0.402970i \(-0.132022\pi\)
−0.666063 + 0.745895i \(0.732022\pi\)
\(90\) 0 0
\(91\) 2.73001 − 3.75754i 0.286183 − 0.393897i
\(92\) 0 0
\(93\) −0.970704 − 0.970704i −0.100657 − 0.100657i
\(94\) 0 0
\(95\) 4.38508 + 9.16927i 0.449900 + 0.940748i
\(96\) 0 0
\(97\) −2.03227 − 3.98855i −0.206346 − 0.404976i 0.764520 − 0.644600i \(-0.222976\pi\)
−0.970866 + 0.239624i \(0.922976\pi\)
\(98\) 0 0
\(99\) 1.40599 0.141308
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 800.2.bq.b.223.3 yes 56
4.3 odd 2 800.2.bq.a.223.5 ✓ 56
25.12 odd 20 800.2.bq.a.287.5 yes 56
100.87 even 20 inner 800.2.bq.b.287.3 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.bq.a.223.5 ✓ 56 4.3 odd 2
800.2.bq.a.287.5 yes 56 25.12 odd 20
800.2.bq.b.223.3 yes 56 1.1 even 1 trivial
800.2.bq.b.287.3 yes 56 100.87 even 20 inner