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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [513,2,Mod(116,513)] 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("513.116"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.cd (of order \(18\), degree \(6\), not 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: \(4.09632562369\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 395.8
Character \(\chi\) \(=\) 513.395
Dual form 513.2.cd.a.413.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.114947 - 0.651899i) q^{2} +(1.46763 - 0.534172i) q^{4} +(2.11716 - 0.373313i) q^{5} +4.63689 q^{7} +(-1.17888 - 2.04188i) q^{8} +(-0.486725 - 1.33727i) q^{10} +(-2.06942 - 1.19478i) q^{11} +(-1.12514 + 1.34089i) q^{13} +(-0.532998 - 3.02278i) q^{14} +(1.19725 - 1.00461i) q^{16} +(-1.52972 + 4.20288i) q^{17} +(-3.09784 + 3.06650i) q^{19} +(2.90779 - 1.67881i) q^{20} +(-0.541001 + 1.48639i) q^{22} +(1.97169 + 5.41717i) q^{23} +(-0.355445 + 0.129372i) q^{25} +(1.00346 + 0.579348i) q^{26} +(6.80521 - 2.47690i) q^{28} +(-5.01985 - 4.21215i) q^{29} +(-2.62988 + 1.51836i) q^{31} +(-4.40483 - 3.69609i) q^{32} +(2.91569 + 0.514115i) q^{34} +(9.81705 - 1.73101i) q^{35} -6.49789i q^{37} +(2.35514 + 1.66699i) q^{38} +(-3.25815 - 3.88291i) q^{40} +(0.835095 + 0.303950i) q^{41} +(-10.1747 - 3.70329i) q^{43} +(-3.67535 - 0.648063i) q^{44} +(3.30481 - 1.90803i) q^{46} +(5.36369 - 6.39219i) q^{47} +14.5007 q^{49} +(0.125195 + 0.216844i) q^{50} +(-0.935021 + 2.56895i) q^{52} +(5.26690 + 4.41945i) q^{53} +(-4.82732 - 1.75700i) q^{55} +(-5.46634 - 9.46798i) q^{56} +(-2.16888 + 3.75661i) q^{58} +(-4.79130 + 4.02038i) q^{59} +(0.265379 - 1.50504i) q^{61} +(1.29212 + 1.53989i) q^{62} +(-0.340260 + 0.589347i) q^{64} +(-1.88154 + 3.25892i) q^{65} +(-2.99917 - 0.528834i) q^{67} +6.98539i q^{68} +(-2.25689 - 6.20075i) q^{70} +(7.57102 - 6.35284i) q^{71} +(1.89766 + 10.7622i) q^{73} +(-4.23597 + 0.746916i) q^{74} +(-2.90843 + 6.15525i) q^{76} +(-9.59565 - 5.54005i) q^{77} +(-1.38770 - 1.65380i) q^{79} +(2.15973 - 2.57387i) q^{80} +(0.102153 - 0.579336i) q^{82} -2.14060i q^{83} +(-1.66969 + 9.46926i) q^{85} +(-1.24462 + 7.05857i) q^{86} +5.63401i q^{88} +(-2.28391 + 12.9527i) q^{89} +(-5.21716 + 6.21757i) q^{91} +(5.78740 + 6.89715i) q^{92} +(-4.78361 - 2.76182i) q^{94} +(-5.41386 + 7.64874i) q^{95} +(1.81563 - 0.320145i) q^{97} +(-1.66682 - 9.45300i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 9 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} - 12 q^{10} + 9 q^{11} - 6 q^{13} + 9 q^{14} + 9 q^{16} + 27 q^{17} - 15 q^{19} + 18 q^{20} - 6 q^{22} + 9 q^{23} - 3 q^{25} - 72 q^{26} - 36 q^{28} - 18 q^{29}+ \cdots - 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.114947 − 0.651899i −0.0812801 − 0.460962i −0.998097 − 0.0616556i \(-0.980362\pi\)
0.916817 − 0.399307i \(-0.130749\pi\)
\(3\) 0 0
\(4\) 1.46763 − 0.534172i 0.733813 − 0.267086i
\(5\) 2.11716 − 0.373313i 0.946824 − 0.166951i 0.321144 − 0.947030i \(-0.395933\pi\)
0.625680 + 0.780080i \(0.284822\pi\)
\(6\) 0 0
\(7\) 4.63689 1.75258 0.876289 − 0.481786i \(-0.160012\pi\)
0.876289 + 0.481786i \(0.160012\pi\)
\(8\) −1.17888 − 2.04188i −0.416798 − 0.721915i
\(9\) 0 0
\(10\) −0.486725 − 1.33727i −0.153916 − 0.422880i
\(11\) −2.06942 − 1.19478i −0.623953 − 0.360239i 0.154454 − 0.988000i \(-0.450638\pi\)
−0.778406 + 0.627761i \(0.783972\pi\)
\(12\) 0 0
\(13\) −1.12514 + 1.34089i −0.312059 + 0.371897i −0.899163 − 0.437614i \(-0.855823\pi\)
0.587104 + 0.809511i \(0.300268\pi\)
\(14\) −0.532998 − 3.02278i −0.142450 − 0.807872i
\(15\) 0 0
\(16\) 1.19725 − 1.00461i 0.299312 − 0.251152i
\(17\) −1.52972 + 4.20288i −0.371013 + 1.01935i 0.603958 + 0.797016i \(0.293589\pi\)
−0.974971 + 0.222333i \(0.928633\pi\)
\(18\) 0 0
\(19\) −3.09784 + 3.06650i −0.710693 + 0.703503i
\(20\) 2.90779 − 1.67881i 0.650202 − 0.375394i
\(21\) 0 0
\(22\) −0.541001 + 1.48639i −0.115342 + 0.316899i
\(23\) 1.97169 + 5.41717i 0.411125 + 1.12956i 0.956593 + 0.291426i \(0.0941296\pi\)
−0.545468 + 0.838132i \(0.683648\pi\)
\(24\) 0 0
\(25\) −0.355445 + 0.129372i −0.0710891 + 0.0258743i
\(26\) 1.00346 + 0.579348i 0.196795 + 0.113619i
\(27\) 0 0
\(28\) 6.80521 − 2.47690i 1.28606 − 0.468089i
\(29\) −5.01985 − 4.21215i −0.932162 − 0.782177i 0.0440421 − 0.999030i \(-0.485976\pi\)
−0.976204 + 0.216853i \(0.930421\pi\)
\(30\) 0 0
\(31\) −2.62988 + 1.51836i −0.472341 + 0.272706i −0.717219 − 0.696848i \(-0.754585\pi\)
0.244878 + 0.969554i \(0.421252\pi\)
\(32\) −4.40483 − 3.69609i −0.778671 − 0.653382i
\(33\) 0 0
\(34\) 2.91569 + 0.514115i 0.500037 + 0.0881701i
\(35\) 9.81705 − 1.73101i 1.65938 − 0.292594i
\(36\) 0 0
\(37\) − 6.49789i − 1.06825i −0.845407 − 0.534123i \(-0.820642\pi\)
0.845407 − 0.534123i \(-0.179358\pi\)
\(38\) 2.35514 + 1.66699i 0.382053 + 0.270422i
\(39\) 0 0
\(40\) −3.25815 − 3.88291i −0.515158 − 0.613942i
\(41\) 0.835095 + 0.303950i 0.130420 + 0.0474690i 0.406406 − 0.913693i \(-0.366782\pi\)
−0.275986 + 0.961162i \(0.589004\pi\)
\(42\) 0 0
\(43\) −10.1747 − 3.70329i −1.55163 − 0.564747i −0.582831 − 0.812594i \(-0.698055\pi\)
−0.968799 + 0.247847i \(0.920277\pi\)
\(44\) −3.67535 − 0.648063i −0.554080 − 0.0976992i
\(45\) 0 0
\(46\) 3.30481 − 1.90803i 0.487267 − 0.281324i
\(47\) 5.36369 − 6.39219i 0.782374 − 0.932397i −0.216664 − 0.976246i \(-0.569518\pi\)
0.999038 + 0.0438491i \(0.0139621\pi\)
\(48\) 0 0
\(49\) 14.5007 2.07153
\(50\) 0.125195 + 0.216844i 0.0177052 + 0.0306663i
\(51\) 0 0
\(52\) −0.935021 + 2.56895i −0.129664 + 0.356249i
\(53\) 5.26690 + 4.41945i 0.723464 + 0.607059i 0.928341 − 0.371729i \(-0.121235\pi\)
−0.204877 + 0.978788i \(0.565679\pi\)
\(54\) 0 0
\(55\) −4.82732 − 1.75700i −0.650916 − 0.236914i
\(56\) −5.46634 − 9.46798i −0.730470 − 1.26521i
\(57\) 0 0
\(58\) −2.16888 + 3.75661i −0.284788 + 0.493267i
\(59\) −4.79130 + 4.02038i −0.623774 + 0.523409i −0.898987 − 0.437974i \(-0.855696\pi\)
0.275213 + 0.961383i \(0.411252\pi\)
\(60\) 0 0
\(61\) 0.265379 − 1.50504i 0.0339783 − 0.192700i −0.963094 − 0.269165i \(-0.913252\pi\)
0.997072 + 0.0764649i \(0.0243633\pi\)
\(62\) 1.29212 + 1.53989i 0.164099 + 0.195566i
\(63\) 0 0
\(64\) −0.340260 + 0.589347i −0.0425324 + 0.0736683i
\(65\) −1.88154 + 3.25892i −0.233376 + 0.404219i
\(66\) 0 0
\(67\) −2.99917 − 0.528834i −0.366407 − 0.0646074i −0.0125864 − 0.999921i \(-0.504006\pi\)
−0.353820 + 0.935313i \(0.615118\pi\)
\(68\) 6.98539i 0.847104i
\(69\) 0 0
\(70\) −2.25689 − 6.20075i −0.269750 − 0.741131i
\(71\) 7.57102 − 6.35284i 0.898515 − 0.753943i −0.0713849 − 0.997449i \(-0.522742\pi\)
0.969899 + 0.243506i \(0.0782974\pi\)
\(72\) 0 0
\(73\) 1.89766 + 10.7622i 0.222104 + 1.25962i 0.868144 + 0.496312i \(0.165313\pi\)
−0.646040 + 0.763303i \(0.723576\pi\)
\(74\) −4.23597 + 0.746916i −0.492421 + 0.0868272i
\(75\) 0 0
\(76\) −2.90843 + 6.15525i −0.333620 + 0.706055i
\(77\) −9.59565 − 5.54005i −1.09353 − 0.631348i
\(78\) 0 0
\(79\) −1.38770 − 1.65380i −0.156129 − 0.186067i 0.682310 − 0.731063i \(-0.260976\pi\)
−0.838438 + 0.544996i \(0.816531\pi\)
\(80\) 2.15973 − 2.57387i 0.241465 − 0.287767i
\(81\) 0 0
\(82\) 0.102153 − 0.579336i 0.0112809 − 0.0639770i
\(83\) − 2.14060i − 0.234962i −0.993075 − 0.117481i \(-0.962518\pi\)
0.993075 − 0.117481i \(-0.0374819\pi\)
\(84\) 0 0
\(85\) −1.66969 + 9.46926i −0.181103 + 1.02708i
\(86\) −1.24462 + 7.05857i −0.134211 + 0.761146i
\(87\) 0 0
\(88\) 5.63401i 0.600588i
\(89\) −2.28391 + 12.9527i −0.242094 + 1.37298i 0.585052 + 0.810996i \(0.301074\pi\)
−0.827146 + 0.561987i \(0.810037\pi\)
\(90\) 0 0
\(91\) −5.21716 + 6.21757i −0.546907 + 0.651779i
\(92\) 5.78740 + 6.89715i 0.603378 + 0.719078i
\(93\) 0 0
\(94\) −4.78361 − 2.76182i −0.493391 − 0.284860i
\(95\) −5.41386 + 7.64874i −0.555451 + 0.784744i
\(96\) 0 0
\(97\) 1.81563 − 0.320145i 0.184350 − 0.0325058i −0.0807110 − 0.996738i \(-0.525719\pi\)
0.265061 + 0.964232i \(0.414608\pi\)
\(98\) −1.66682 − 9.45300i −0.168374 − 0.954897i
\(99\) 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.cd.a.395.8 108
3.2 odd 2 171.2.bd.a.167.11 yes 108
9.2 odd 6 513.2.bo.a.224.11 108
9.7 even 3 171.2.x.a.110.8 yes 108
19.14 odd 18 513.2.bo.a.71.11 108
57.14 even 18 171.2.x.a.14.8 ✓ 108
171.52 odd 18 171.2.bd.a.128.11 yes 108
171.128 even 18 inner 513.2.cd.a.413.8 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.x.a.14.8 ✓ 108 57.14 even 18
171.2.x.a.110.8 yes 108 9.7 even 3
171.2.bd.a.128.11 yes 108 171.52 odd 18
171.2.bd.a.167.11 yes 108 3.2 odd 2
513.2.bo.a.71.11 108 19.14 odd 18
513.2.bo.a.224.11 108 9.2 odd 6
513.2.cd.a.395.8 108 1.1 even 1 trivial
513.2.cd.a.413.8 108 171.128 even 18 inner