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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [513,2,Mod(107,513)] 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("513.107"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,4,0,0,8] 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: \(4.09632562369\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-14})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 14x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 107.2
Root \(3.24037 + 1.87083i\) of defining polynomial
Character \(\chi\) \(=\) 513.107
Dual form 513.2.m.d.350.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{4} +(3.24037 - 1.87083i) q^{5} +2.00000 q^{7} +3.74166i q^{11} +(4.50000 + 2.59808i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(-6.48074 + 3.74166i) q^{17} +(-3.50000 - 2.59808i) q^{19} -7.48331i q^{20} +(-3.24037 - 1.87083i) q^{23} +(4.50000 - 7.79423i) q^{25} +(2.00000 - 3.46410i) q^{28} +(-3.24037 + 5.61249i) q^{29} +1.73205i q^{31} +(6.48074 - 3.74166i) q^{35} +5.19615i q^{37} +(3.24037 + 5.61249i) q^{41} +(-2.50000 - 4.33013i) q^{43} +(6.48074 + 3.74166i) q^{44} +(-3.24037 - 1.87083i) q^{47} -3.00000 q^{49} +(9.00000 - 5.19615i) q^{52} +(3.24037 - 5.61249i) q^{53} +(7.00000 + 12.1244i) q^{55} +(-6.48074 - 11.2250i) q^{59} +(3.50000 - 6.06218i) q^{61} -8.00000 q^{64} +19.4422 q^{65} +14.9666i q^{68} +(-3.24037 - 5.61249i) q^{71} +(2.00000 + 3.46410i) q^{73} +(-8.00000 + 3.46410i) q^{76} +7.48331i q^{77} +(-4.50000 + 2.59808i) q^{79} +(-12.9615 - 7.48331i) q^{80} -7.48331i q^{83} +(-14.0000 + 24.2487i) q^{85} +(-6.48074 + 11.2250i) q^{89} +(9.00000 + 5.19615i) q^{91} +(-6.48074 + 3.74166i) q^{92} +(-16.2019 - 1.87083i) q^{95} +(4.50000 - 2.59808i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 8 q^{7} + 18 q^{13} - 8 q^{16} - 14 q^{19} + 18 q^{25} + 8 q^{28} - 10 q^{43} - 12 q^{49} + 36 q^{52} + 28 q^{55} + 14 q^{61} - 32 q^{64} + 8 q^{73} - 32 q^{76} - 18 q^{79} - 56 q^{85} + 36 q^{91}+ \cdots + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/513\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 0 0
\(4\) 1.00000 − 1.73205i 0.500000 − 0.866025i
\(5\) 3.24037 − 1.87083i 1.44914 − 0.836660i 0.450708 − 0.892672i \(-0.351172\pi\)
0.998430 + 0.0560116i \(0.0178384\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.74166i 1.12815i 0.825723 + 0.564076i \(0.190768\pi\)
−0.825723 + 0.564076i \(0.809232\pi\)
\(12\) 0 0
\(13\) 4.50000 + 2.59808i 1.24808 + 0.720577i 0.970725 − 0.240192i \(-0.0772105\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 − 3.46410i −0.500000 − 0.866025i
\(17\) −6.48074 + 3.74166i −1.57181 + 0.907485i −0.575863 + 0.817546i \(0.695334\pi\)
−0.995947 + 0.0899392i \(0.971333\pi\)
\(18\) 0 0
\(19\) −3.50000 − 2.59808i −0.802955 − 0.596040i
\(20\) − 7.48331i − 1.67332i
\(21\) 0 0
\(22\) 0 0
\(23\) −3.24037 − 1.87083i −0.675664 − 0.390095i 0.122555 − 0.992462i \(-0.460891\pi\)
−0.798219 + 0.602367i \(0.794224\pi\)
\(24\) 0 0
\(25\) 4.50000 − 7.79423i 0.900000 − 1.55885i
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 − 3.46410i 0.377964 − 0.654654i
\(29\) −3.24037 + 5.61249i −0.601722 + 1.04221i 0.390839 + 0.920459i \(0.372185\pi\)
−0.992560 + 0.121753i \(0.961148\pi\)
\(30\) 0 0
\(31\) 1.73205i 0.311086i 0.987829 + 0.155543i \(0.0497126\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.48074 − 3.74166i 1.09545 − 0.632456i
\(36\) 0 0
\(37\) 5.19615i 0.854242i 0.904194 + 0.427121i \(0.140472\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.24037 + 5.61249i 0.506061 + 0.876523i 0.999975 + 0.00701264i \(0.00223221\pi\)
−0.493915 + 0.869510i \(0.664434\pi\)
\(42\) 0 0
\(43\) −2.50000 − 4.33013i −0.381246 − 0.660338i 0.609994 − 0.792406i \(-0.291172\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 6.48074 + 3.74166i 0.977008 + 0.564076i
\(45\) 0 0
\(46\) 0 0
\(47\) −3.24037 − 1.87083i −0.472657 − 0.272888i 0.244695 − 0.969600i \(-0.421312\pi\)
−0.717351 + 0.696712i \(0.754646\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 9.00000 − 5.19615i 1.24808 − 0.720577i
\(53\) 3.24037 − 5.61249i 0.445099 − 0.770934i −0.552960 − 0.833208i \(-0.686502\pi\)
0.998059 + 0.0622735i \(0.0198351\pi\)
\(54\) 0 0
\(55\) 7.00000 + 12.1244i 0.943880 + 1.63485i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.48074 − 11.2250i −0.843721 − 1.46137i −0.886728 − 0.462292i \(-0.847027\pi\)
0.0430071 − 0.999075i \(-0.486306\pi\)
\(60\) 0 0
\(61\) 3.50000 − 6.06218i 0.448129 − 0.776182i −0.550135 − 0.835076i \(-0.685424\pi\)
0.998264 + 0.0588933i \(0.0187572\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 19.4422 2.41151
\(66\) 0 0
\(67\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 14.9666i 1.81497i
\(69\) 0 0
\(70\) 0 0
\(71\) −3.24037 − 5.61249i −0.384561 − 0.666080i 0.607147 − 0.794590i \(-0.292314\pi\)
−0.991708 + 0.128510i \(0.958981\pi\)
\(72\) 0 0
\(73\) 2.00000 + 3.46410i 0.234082 + 0.405442i 0.959006 − 0.283387i \(-0.0914581\pi\)
−0.724923 + 0.688830i \(0.758125\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −8.00000 + 3.46410i −0.917663 + 0.397360i
\(77\) 7.48331i 0.852803i
\(78\) 0 0
\(79\) −4.50000 + 2.59808i −0.506290 + 0.292306i −0.731307 − 0.682048i \(-0.761089\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −12.9615 − 7.48331i −1.44914 − 0.836660i
\(81\) 0 0
\(82\) 0 0
\(83\) − 7.48331i − 0.821401i −0.911770 − 0.410700i \(-0.865284\pi\)
0.911770 − 0.410700i \(-0.134716\pi\)
\(84\) 0 0
\(85\) −14.0000 + 24.2487i −1.51851 + 2.63014i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.48074 + 11.2250i −0.686957 + 1.18984i 0.285860 + 0.958271i \(0.407721\pi\)
−0.972817 + 0.231573i \(0.925613\pi\)
\(90\) 0 0
\(91\) 9.00000 + 5.19615i 0.943456 + 0.544705i
\(92\) −6.48074 + 3.74166i −0.675664 + 0.390095i
\(93\) 0 0
\(94\) 0 0
\(95\) −16.2019 − 1.87083i −1.66227 − 0.191943i
\(96\) 0 0
\(97\) 4.50000 − 2.59808i 0.456906 − 0.263795i −0.253837 − 0.967247i \(-0.581693\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 0 0
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 513.2.m.d.107.2 yes 4
3.2 odd 2 inner 513.2.m.d.107.1 ✓ 4
19.8 odd 6 inner 513.2.m.d.350.1 yes 4
57.8 even 6 inner 513.2.m.d.350.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
513.2.m.d.107.1 ✓ 4 3.2 odd 2 inner
513.2.m.d.107.2 yes 4 1.1 even 1 trivial
513.2.m.d.350.1 yes 4 19.8 odd 6 inner
513.2.m.d.350.2 yes 4 57.8 even 6 inner