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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [736,2,Mod(49,736)] 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("736.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 11, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 736.x (of order \(22\), degree \(10\), 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: \(5.87698958877\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 49.9
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.496871 - 0.226913i) q^{3} +(1.16498 + 1.00946i) q^{5} +(-0.981244 + 0.630607i) q^{7} +(-1.76919 - 2.04176i) q^{9} +(-4.28986 - 0.616789i) q^{11} +(-1.76371 + 2.74438i) q^{13} +(-0.349784 - 0.765920i) q^{15} +(-3.79307 - 1.11375i) q^{17} +(-0.469533 - 1.59908i) q^{19} +(0.630645 - 0.0906731i) q^{21} +(1.22189 - 4.63756i) q^{23} +(-0.373409 - 2.59712i) q^{25} +(0.877433 + 2.98826i) q^{27} +(-1.96335 + 6.68655i) q^{29} +(-1.28600 - 2.81594i) q^{31} +(1.99155 + 1.27989i) q^{33} +(-1.77970 - 0.255882i) q^{35} +(-2.91860 + 2.52898i) q^{37} +(1.49907 - 0.963395i) q^{39} +(4.80593 - 5.54634i) q^{41} +(0.605421 + 0.276487i) q^{43} -4.16452i q^{45} -8.53098 q^{47} +(-2.34273 + 5.12986i) q^{49} +(1.63194 + 1.41409i) q^{51} +(-5.72300 - 8.90516i) q^{53} +(-4.37497 - 5.04898i) q^{55} +(-0.129556 + 0.901082i) q^{57} +(-0.863541 + 1.34370i) q^{59} +(-1.81729 + 0.829930i) q^{61} +(3.02355 + 0.887795i) q^{63} +(-4.82501 + 1.41675i) q^{65} +(2.22718 - 0.320221i) q^{67} +(-1.65945 + 2.02701i) q^{69} +(1.69240 + 11.7709i) q^{71} +(-6.66904 + 1.95821i) q^{73} +(-0.403785 + 1.37516i) q^{75} +(4.59835 - 2.10000i) q^{77} +(4.03633 + 2.59399i) q^{79} +(-0.911340 + 6.33851i) q^{81} +(-13.6936 + 11.8656i) q^{83} +(-3.29456 - 5.12644i) q^{85} +(2.49280 - 2.87684i) q^{87} +(7.74096 - 16.9503i) q^{89} -3.80511i q^{91} +1.69097i q^{93} +(1.06721 - 2.33687i) q^{95} +(-0.978945 + 1.12976i) q^{97} +(6.33025 + 9.85007i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/736\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(415\) \(645\)
\(\chi(n)\) \(e\left(\frac{8}{11}\right)\) \(1\) \(-1\)

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
\(3\) −0.496871 − 0.226913i −0.286869 − 0.131009i 0.266784 − 0.963756i \(-0.414039\pi\)
−0.553652 + 0.832748i \(0.686766\pi\)
\(4\) 0 0
\(5\) 1.16498 + 1.00946i 0.520993 + 0.451443i 0.875226 − 0.483714i \(-0.160712\pi\)
−0.354233 + 0.935157i \(0.615258\pi\)
\(6\) 0 0
\(7\) −0.981244 + 0.630607i −0.370875 + 0.238347i −0.712780 − 0.701387i \(-0.752564\pi\)
0.341905 + 0.939735i \(0.388928\pi\)
\(8\) 0 0
\(9\) −1.76919 − 2.04176i −0.589730 − 0.680585i
\(10\) 0 0
\(11\) −4.28986 − 0.616789i −1.29344 − 0.185969i −0.538984 − 0.842316i \(-0.681192\pi\)
−0.754459 + 0.656347i \(0.772101\pi\)
\(12\) 0 0
\(13\) −1.76371 + 2.74438i −0.489164 + 0.761154i −0.994828 − 0.101577i \(-0.967611\pi\)
0.505664 + 0.862731i \(0.331248\pi\)
\(14\) 0 0
\(15\) −0.349784 − 0.765920i −0.0903138 − 0.197760i
\(16\) 0 0
\(17\) −3.79307 − 1.11375i −0.919955 − 0.270123i −0.212730 − 0.977111i \(-0.568235\pi\)
−0.707226 + 0.706988i \(0.750054\pi\)
\(18\) 0 0
\(19\) −0.469533 − 1.59908i −0.107718 − 0.366855i 0.887938 − 0.459964i \(-0.152138\pi\)
−0.995656 + 0.0931091i \(0.970319\pi\)
\(20\) 0 0
\(21\) 0.630645 − 0.0906731i 0.137618 − 0.0197865i
\(22\) 0 0
\(23\) 1.22189 − 4.63756i 0.254782 − 0.966998i
\(24\) 0 0
\(25\) −0.373409 − 2.59712i −0.0746818 − 0.519423i
\(26\) 0 0
\(27\) 0.877433 + 2.98826i 0.168862 + 0.575092i
\(28\) 0 0
\(29\) −1.96335 + 6.68655i −0.364584 + 1.24166i 0.549279 + 0.835639i \(0.314902\pi\)
−0.913863 + 0.406022i \(0.866916\pi\)
\(30\) 0 0
\(31\) −1.28600 − 2.81594i −0.230972 − 0.505759i 0.758289 − 0.651919i \(-0.226036\pi\)
−0.989261 + 0.146160i \(0.953308\pi\)
\(32\) 0 0
\(33\) 1.99155 + 1.27989i 0.346685 + 0.222801i
\(34\) 0 0
\(35\) −1.77970 − 0.255882i −0.300824 − 0.0432519i
\(36\) 0 0
\(37\) −2.91860 + 2.52898i −0.479814 + 0.415762i −0.860897 − 0.508779i \(-0.830097\pi\)
0.381082 + 0.924541i \(0.375551\pi\)
\(38\) 0 0
\(39\) 1.49907 − 0.963395i 0.240044 − 0.154267i
\(40\) 0 0
\(41\) 4.80593 − 5.54634i 0.750560 − 0.866192i −0.244062 − 0.969760i \(-0.578480\pi\)
0.994622 + 0.103567i \(0.0330257\pi\)
\(42\) 0 0
\(43\) 0.605421 + 0.276487i 0.0923258 + 0.0421638i 0.461043 − 0.887378i \(-0.347476\pi\)
−0.368717 + 0.929542i \(0.620203\pi\)
\(44\) 0 0
\(45\) − 4.16452i − 0.620810i
\(46\) 0 0
\(47\) −8.53098 −1.24437 −0.622185 − 0.782870i \(-0.713755\pi\)
−0.622185 + 0.782870i \(0.713755\pi\)
\(48\) 0 0
\(49\) −2.34273 + 5.12986i −0.334676 + 0.732838i
\(50\) 0 0
\(51\) 1.63194 + 1.41409i 0.228518 + 0.198012i
\(52\) 0 0
\(53\) −5.72300 − 8.90516i −0.786114 − 1.22322i −0.970675 − 0.240396i \(-0.922723\pi\)
0.184561 − 0.982821i \(-0.440914\pi\)
\(54\) 0 0
\(55\) −4.37497 − 5.04898i −0.589921 − 0.680805i
\(56\) 0 0
\(57\) −0.129556 + 0.901082i −0.0171601 + 0.119351i
\(58\) 0 0
\(59\) −0.863541 + 1.34370i −0.112423 + 0.174934i −0.892906 − 0.450243i \(-0.851337\pi\)
0.780482 + 0.625178i \(0.214974\pi\)
\(60\) 0 0
\(61\) −1.81729 + 0.829930i −0.232681 + 0.106262i −0.528344 − 0.849031i \(-0.677187\pi\)
0.295663 + 0.955292i \(0.404459\pi\)
\(62\) 0 0
\(63\) 3.02355 + 0.887795i 0.380932 + 0.111852i
\(64\) 0 0
\(65\) −4.82501 + 1.41675i −0.598469 + 0.175726i
\(66\) 0 0
\(67\) 2.22718 − 0.320221i 0.272094 − 0.0391212i −0.00491729 − 0.999988i \(-0.501565\pi\)
0.277011 + 0.960867i \(0.410656\pi\)
\(68\) 0 0
\(69\) −1.65945 + 2.02701i −0.199774 + 0.244023i
\(70\) 0 0
\(71\) 1.69240 + 11.7709i 0.200851 + 1.39695i 0.801766 + 0.597638i \(0.203894\pi\)
−0.600915 + 0.799313i \(0.705197\pi\)
\(72\) 0 0
\(73\) −6.66904 + 1.95821i −0.780552 + 0.229191i −0.647650 − 0.761938i \(-0.724248\pi\)
−0.132903 + 0.991129i \(0.542430\pi\)
\(74\) 0 0
\(75\) −0.403785 + 1.37516i −0.0466250 + 0.158790i
\(76\) 0 0
\(77\) 4.59835 − 2.10000i 0.524031 − 0.239317i
\(78\) 0 0
\(79\) 4.03633 + 2.59399i 0.454123 + 0.291847i 0.747637 − 0.664108i \(-0.231188\pi\)
−0.293514 + 0.955955i \(0.594825\pi\)
\(80\) 0 0
\(81\) −0.911340 + 6.33851i −0.101260 + 0.704279i
\(82\) 0 0
\(83\) −13.6936 + 11.8656i −1.50307 + 1.30242i −0.682462 + 0.730921i \(0.739091\pi\)
−0.820606 + 0.571495i \(0.806364\pi\)
\(84\) 0 0
\(85\) −3.29456 − 5.12644i −0.357345 − 0.556040i
\(86\) 0 0
\(87\) 2.49280 − 2.87684i 0.267256 − 0.308430i
\(88\) 0 0
\(89\) 7.74096 − 16.9503i 0.820540 − 1.79673i 0.267552 − 0.963543i \(-0.413785\pi\)
0.552988 − 0.833189i \(-0.313487\pi\)
\(90\) 0 0
\(91\) − 3.80511i − 0.398884i
\(92\) 0 0
\(93\) 1.69097i 0.175346i
\(94\) 0 0
\(95\) 1.06721 − 2.33687i 0.109494 − 0.239758i
\(96\) 0 0
\(97\) −0.978945 + 1.12976i −0.0993968 + 0.114710i −0.803268 − 0.595617i \(-0.796908\pi\)
0.703872 + 0.710327i \(0.251453\pi\)
\(98\) 0 0
\(99\) 6.33025 + 9.85007i 0.636215 + 0.989969i
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 736.2.x.a.49.9 220
4.3 odd 2 184.2.p.a.141.16 yes 220
8.3 odd 2 184.2.p.a.141.5 yes 220
8.5 even 2 inner 736.2.x.a.49.14 220
23.8 even 11 inner 736.2.x.a.721.14 220
92.31 odd 22 184.2.p.a.77.5 ✓ 220
184.77 even 22 inner 736.2.x.a.721.9 220
184.123 odd 22 184.2.p.a.77.16 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.5 ✓ 220 92.31 odd 22
184.2.p.a.77.16 yes 220 184.123 odd 22
184.2.p.a.141.5 yes 220 8.3 odd 2
184.2.p.a.141.16 yes 220 4.3 odd 2
736.2.x.a.49.9 220 1.1 even 1 trivial
736.2.x.a.49.14 220 8.5 even 2 inner
736.2.x.a.721.9 220 184.77 even 22 inner
736.2.x.a.721.14 220 23.8 even 11 inner