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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 223.4
Character \(\chi\) \(=\) 800.223
Dual form 800.2.bq.a.287.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.0719261 - 0.141163i) q^{3} +(-0.475770 + 2.18487i) q^{5} +(2.98667 + 2.98667i) q^{7} +(1.74860 - 2.40674i) q^{9} +(-1.53022 - 2.10616i) q^{11} +(-0.442022 - 2.79082i) q^{13} +(0.342642 - 0.0899879i) q^{15} +(5.14717 + 2.62261i) q^{17} +(0.539586 + 1.66067i) q^{19} +(0.206788 - 0.636427i) q^{21} +(-1.32955 + 8.39445i) q^{23} +(-4.54729 - 2.07899i) q^{25} +(-0.934954 - 0.148082i) q^{27} +(7.60053 + 2.46956i) q^{29} +(-4.78787 + 1.55567i) q^{31} +(-0.187249 + 0.367498i) q^{33} +(-7.94645 + 5.10451i) q^{35} +(-3.15191 + 0.499213i) q^{37} +(-0.362167 + 0.263130i) q^{39} +(1.17976 + 0.857147i) q^{41} +(-5.27067 + 5.27067i) q^{43} +(4.42648 + 4.96552i) q^{45} +(8.06293 - 4.10827i) q^{47} +10.8404i q^{49} -0.915223i q^{51} +(11.8565 - 6.04120i) q^{53} +(5.32971 - 2.34127i) q^{55} +(0.195615 - 0.195615i) q^{57} +(-3.70605 - 2.69261i) q^{59} +(-2.83917 + 2.06278i) q^{61} +(12.4107 - 1.96565i) q^{63} +(6.30786 + 0.362026i) q^{65} +(-1.15082 + 2.25861i) q^{67} +(1.28061 - 0.416097i) q^{69} +(5.42288 + 1.76200i) q^{71} +(3.08081 + 0.487953i) q^{73} +(0.0335928 + 0.791441i) q^{75} +(1.72016 - 10.8607i) q^{77} +(-2.59532 + 7.98759i) q^{79} +(-2.71154 - 8.34526i) q^{81} +(7.08348 + 3.60921i) q^{83} +(-8.17893 + 9.99812i) q^{85} +(-0.198066 - 1.25054i) q^{87} +(-2.13580 - 2.93967i) q^{89} +(7.01508 - 9.65543i) q^{91} +(0.563976 + 0.563976i) q^{93} +(-3.88507 + 0.388825i) q^{95} +(-6.94898 - 13.6382i) q^{97} -7.74473 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{5} - 4 q^{7} - 6 q^{13} - 42 q^{15} - 2 q^{17} - 18 q^{19} + 16 q^{21} - 8 q^{23} - 34 q^{25} + 42 q^{27} + 20 q^{31} - 36 q^{33} + 22 q^{35} + 20 q^{37} - 36 q^{39} + 16 q^{41} - 32 q^{43}+ \cdots + 132 q^{99}+O(q^{100}) \) 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
\(3\) −0.0719261 − 0.141163i −0.0415265 − 0.0815004i 0.869311 − 0.494266i \(-0.164563\pi\)
−0.910837 + 0.412765i \(0.864563\pi\)
\(4\) 0 0
\(5\) −0.475770 + 2.18487i −0.212771 + 0.977102i
\(6\) 0 0
\(7\) 2.98667 + 2.98667i 1.12886 + 1.12886i 0.990364 + 0.138492i \(0.0442256\pi\)
0.138492 + 0.990364i \(0.455774\pi\)
\(8\) 0 0
\(9\) 1.74860 − 2.40674i 0.582867 − 0.802248i
\(10\) 0 0
\(11\) −1.53022 − 2.10616i −0.461377 − 0.635031i 0.513416 − 0.858140i \(-0.328380\pi\)
−0.974794 + 0.223108i \(0.928380\pi\)
\(12\) 0 0
\(13\) −0.442022 − 2.79082i −0.122595 − 0.774033i −0.970003 − 0.243092i \(-0.921838\pi\)
0.847408 − 0.530942i \(-0.178162\pi\)
\(14\) 0 0
\(15\) 0.342642 − 0.0899879i 0.0884699 − 0.0232348i
\(16\) 0 0
\(17\) 5.14717 + 2.62261i 1.24837 + 0.636077i 0.948159 − 0.317795i \(-0.102943\pi\)
0.300212 + 0.953872i \(0.402943\pi\)
\(18\) 0 0
\(19\) 0.539586 + 1.66067i 0.123789 + 0.380985i 0.993679 − 0.112263i \(-0.0358098\pi\)
−0.869889 + 0.493247i \(0.835810\pi\)
\(20\) 0 0
\(21\) 0.206788 − 0.636427i 0.0451247 − 0.138880i
\(22\) 0 0
\(23\) −1.32955 + 8.39445i −0.277230 + 1.75036i 0.319190 + 0.947691i \(0.396589\pi\)
−0.596420 + 0.802672i \(0.703411\pi\)
\(24\) 0 0
\(25\) −4.54729 − 2.07899i −0.909457 − 0.415797i
\(26\) 0 0
\(27\) −0.934954 − 0.148082i −0.179932 − 0.0284984i
\(28\) 0 0
\(29\) 7.60053 + 2.46956i 1.41138 + 0.458586i 0.912854 − 0.408286i \(-0.133873\pi\)
0.498530 + 0.866873i \(0.333873\pi\)
\(30\) 0 0
\(31\) −4.78787 + 1.55567i −0.859927 + 0.279407i −0.705598 − 0.708612i \(-0.749322\pi\)
−0.154329 + 0.988019i \(0.549322\pi\)
\(32\) 0 0
\(33\) −0.187249 + 0.367498i −0.0325959 + 0.0639731i
\(34\) 0 0
\(35\) −7.94645 + 5.10451i −1.34319 + 0.862820i
\(36\) 0 0
\(37\) −3.15191 + 0.499213i −0.518171 + 0.0820701i −0.410042 − 0.912067i \(-0.634486\pi\)
−0.108129 + 0.994137i \(0.534486\pi\)
\(38\) 0 0
\(39\) −0.362167 + 0.263130i −0.0579931 + 0.0421345i
\(40\) 0 0
\(41\) 1.17976 + 0.857147i 0.184248 + 0.133864i 0.676086 − 0.736823i \(-0.263675\pi\)
−0.491838 + 0.870687i \(0.663675\pi\)
\(42\) 0 0
\(43\) −5.27067 + 5.27067i −0.803769 + 0.803769i −0.983682 − 0.179913i \(-0.942418\pi\)
0.179913 + 0.983682i \(0.442418\pi\)
\(44\) 0 0
\(45\) 4.42648 + 4.96552i 0.659861 + 0.740216i
\(46\) 0 0
\(47\) 8.06293 − 4.10827i 1.17610 − 0.599253i 0.246976 − 0.969022i \(-0.420563\pi\)
0.929124 + 0.369769i \(0.120563\pi\)
\(48\) 0 0
\(49\) 10.8404i 1.54863i
\(50\) 0 0
\(51\) − 0.915223i − 0.128157i
\(52\) 0 0
\(53\) 11.8565 − 6.04120i 1.62862 − 0.829823i 0.630038 − 0.776564i \(-0.283039\pi\)
0.998581 − 0.0532587i \(-0.0169608\pi\)
\(54\) 0 0
\(55\) 5.32971 − 2.34127i 0.718658 − 0.315697i
\(56\) 0 0
\(57\) 0.195615 − 0.195615i 0.0259099 − 0.0259099i
\(58\) 0 0
\(59\) −3.70605 − 2.69261i −0.482487 − 0.350547i 0.319801 − 0.947485i \(-0.396384\pi\)
−0.802288 + 0.596937i \(0.796384\pi\)
\(60\) 0 0
\(61\) −2.83917 + 2.06278i −0.363518 + 0.264111i −0.754518 − 0.656279i \(-0.772129\pi\)
0.391000 + 0.920391i \(0.372129\pi\)
\(62\) 0 0
\(63\) 12.4107 − 1.96565i 1.56360 − 0.247649i
\(64\) 0 0
\(65\) 6.30786 + 0.362026i 0.782394 + 0.0449039i
\(66\) 0 0
\(67\) −1.15082 + 2.25861i −0.140595 + 0.275933i −0.950558 − 0.310548i \(-0.899487\pi\)
0.809963 + 0.586482i \(0.199487\pi\)
\(68\) 0 0
\(69\) 1.28061 − 0.416097i 0.154168 − 0.0500921i
\(70\) 0 0
\(71\) 5.42288 + 1.76200i 0.643577 + 0.209111i 0.612580 − 0.790409i \(-0.290132\pi\)
0.0309972 + 0.999519i \(0.490132\pi\)
\(72\) 0 0
\(73\) 3.08081 + 0.487953i 0.360582 + 0.0571106i 0.334097 − 0.942539i \(-0.391569\pi\)
0.0264848 + 0.999649i \(0.491569\pi\)
\(74\) 0 0
\(75\) 0.0335928 + 0.791441i 0.00387897 + 0.0913878i
\(76\) 0 0
\(77\) 1.72016 − 10.8607i 0.196030 − 1.23769i
\(78\) 0 0
\(79\) −2.59532 + 7.98759i −0.291997 + 0.898674i 0.692217 + 0.721690i \(0.256634\pi\)
−0.984214 + 0.176984i \(0.943366\pi\)
\(80\) 0 0
\(81\) −2.71154 − 8.34526i −0.301282 − 0.927251i
\(82\) 0 0
\(83\) 7.08348 + 3.60921i 0.777513 + 0.396163i 0.797255 − 0.603643i \(-0.206285\pi\)
−0.0197423 + 0.999805i \(0.506285\pi\)
\(84\) 0 0
\(85\) −8.17893 + 9.99812i −0.887129 + 1.08445i
\(86\) 0 0
\(87\) −0.198066 − 1.25054i −0.0212349 − 0.134072i
\(88\) 0 0
\(89\) −2.13580 − 2.93967i −0.226394 − 0.311605i 0.680676 − 0.732585i \(-0.261686\pi\)
−0.907070 + 0.420980i \(0.861686\pi\)
\(90\) 0 0
\(91\) 7.01508 − 9.65543i 0.735380 − 1.01216i
\(92\) 0 0
\(93\) 0.563976 + 0.563976i 0.0584816 + 0.0584816i
\(94\) 0 0
\(95\) −3.88507 + 0.388825i −0.398600 + 0.0398926i
\(96\) 0 0
\(97\) −6.94898 − 13.6382i −0.705562 − 1.38474i −0.913596 − 0.406623i \(-0.866706\pi\)
0.208034 − 0.978122i \(-0.433294\pi\)
\(98\) 0 0
\(99\) −7.74473 −0.778375
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.a.223.4 ✓ 56
4.3 odd 2 800.2.bq.b.223.4 yes 56
25.12 odd 20 800.2.bq.b.287.4 yes 56
100.87 even 20 inner 800.2.bq.a.287.4 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.bq.a.223.4 ✓ 56 1.1 even 1 trivial
800.2.bq.a.287.4 yes 56 100.87 even 20 inner
800.2.bq.b.223.4 yes 56 4.3 odd 2
800.2.bq.b.287.4 yes 56 25.12 odd 20