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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [297,2,Mod(37,297)] 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("297.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(297, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([10, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.n (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.37155694003\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 289.6
Character \(\chi\) \(=\) 297.289
Dual form 297.2.n.b.37.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0603978 + 0.574647i) q^{2} +(1.62972 - 0.346409i) q^{4} +(0.314048 - 2.98797i) q^{5} +(-0.779082 - 0.865259i) q^{7} +(0.654602 + 2.01466i) q^{8} +1.73599 q^{10} +(0.565602 - 3.26804i) q^{11} +(-3.60501 - 1.60505i) q^{13} +(0.450163 - 0.499957i) q^{14} +(1.92600 - 0.857509i) q^{16} +(-0.254185 - 0.184677i) q^{17} +(1.96794 + 6.05670i) q^{19} +(-0.523246 - 4.97835i) q^{20} +(1.91213 + 0.127639i) q^{22} +(-0.0427501 - 0.0740453i) q^{23} +(-3.93860 - 0.837174i) q^{25} +(0.704604 - 2.16855i) q^{26} +(-1.56942 - 1.14025i) q^{28} +(5.15218 + 5.72208i) q^{29} +(5.96439 + 2.65552i) q^{31} +(2.72743 + 4.72404i) q^{32} +(0.0907715 - 0.157221i) q^{34} +(-2.83004 + 2.05614i) q^{35} +(-1.92922 + 5.93753i) q^{37} +(-3.36160 + 1.49668i) q^{38} +(6.22531 - 1.32323i) q^{40} +(-3.88703 + 4.31698i) q^{41} +(3.39229 - 5.87562i) q^{43} +(-0.210303 - 5.52194i) q^{44} +(0.0399679 - 0.0290384i) q^{46} +(-0.320574 - 0.0681401i) q^{47} +(0.589996 - 5.61344i) q^{49} +(0.243197 - 2.31387i) q^{50} +(-6.43117 - 1.36699i) q^{52} +(-1.96000 + 1.42402i) q^{53} +(-9.58718 - 2.71632i) q^{55} +(1.23321 - 2.13598i) q^{56} +(-2.97699 + 3.30629i) q^{58} +(-2.24586 + 0.477372i) q^{59} +(-12.5039 + 5.56709i) q^{61} +(-1.16575 + 3.58780i) q^{62} +(0.861321 - 0.625787i) q^{64} +(-5.92799 + 10.2676i) q^{65} +(-5.83989 - 10.1150i) q^{67} +(-0.478226 - 0.212920i) q^{68} +(-1.35248 - 1.50208i) q^{70} +(7.05272 + 5.12410i) q^{71} +(0.910538 - 2.80235i) q^{73} +(-3.52850 - 0.750007i) q^{74} +(5.30529 + 9.18904i) q^{76} +(-3.26835 + 2.05668i) q^{77} +(0.928965 + 8.83851i) q^{79} +(-1.95735 - 6.02412i) q^{80} +(-2.71551 - 1.97293i) q^{82} +(-4.52916 + 2.01651i) q^{83} +(-0.631634 + 0.701501i) q^{85} +(3.58129 + 1.59449i) q^{86} +(6.95423 - 0.999772i) q^{88} -2.12862 q^{89} +(1.41981 + 4.36973i) q^{91} +(-0.0953208 - 0.105864i) q^{92} +(0.0197945 - 0.188332i) q^{94} +(18.7153 - 3.97805i) q^{95} +(0.00928623 + 0.0883526i) q^{97} +3.26138 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + q^{2} + 11 q^{4} + 8 q^{5} - 2 q^{7} - 6 q^{8} - 8 q^{10} + 2 q^{11} - 11 q^{13} + 10 q^{14} - 9 q^{16} + 20 q^{17} + 8 q^{19} + 45 q^{20} - 16 q^{22} - 20 q^{23} + 11 q^{25} + 12 q^{26} - 54 q^{28}+ \cdots + 328 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(244\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0603978 + 0.574647i 0.0427077 + 0.406337i 0.994902 + 0.100844i \(0.0321544\pi\)
−0.952195 + 0.305492i \(0.901179\pi\)
\(3\) 0 0
\(4\) 1.62972 − 0.346409i 0.814862 − 0.173204i
\(5\) 0.314048 − 2.98797i 0.140447 − 1.33626i −0.666442 − 0.745557i \(-0.732184\pi\)
0.806889 − 0.590704i \(-0.201150\pi\)
\(6\) 0 0
\(7\) −0.779082 − 0.865259i −0.294465 − 0.327037i 0.577699 − 0.816250i \(-0.303951\pi\)
−0.872164 + 0.489213i \(0.837284\pi\)
\(8\) 0.654602 + 2.01466i 0.231437 + 0.712289i
\(9\) 0 0
\(10\) 1.73599 0.548970
\(11\) 0.565602 − 3.26804i 0.170535 − 0.985352i
\(12\) 0 0
\(13\) −3.60501 − 1.60505i −0.999849 − 0.445161i −0.159495 − 0.987199i \(-0.550986\pi\)
−0.840354 + 0.542037i \(0.817653\pi\)
\(14\) 0.450163 − 0.499957i 0.120311 − 0.133619i
\(15\) 0 0
\(16\) 1.92600 − 0.857509i 0.481499 − 0.214377i
\(17\) −0.254185 − 0.184677i −0.0616490 − 0.0447906i 0.556534 − 0.830825i \(-0.312131\pi\)
−0.618183 + 0.786034i \(0.712131\pi\)
\(18\) 0 0
\(19\) 1.96794 + 6.05670i 0.451477 + 1.38950i 0.875222 + 0.483721i \(0.160715\pi\)
−0.423746 + 0.905781i \(0.639285\pi\)
\(20\) −0.523246 − 4.97835i −0.117001 − 1.11319i
\(21\) 0 0
\(22\) 1.91213 + 0.127639i 0.407668 + 0.0272126i
\(23\) −0.0427501 − 0.0740453i −0.00891401 − 0.0154395i 0.861534 − 0.507700i \(-0.169504\pi\)
−0.870448 + 0.492260i \(0.836171\pi\)
\(24\) 0 0
\(25\) −3.93860 − 0.837174i −0.787719 − 0.167435i
\(26\) 0.704604 − 2.16855i 0.138184 − 0.425287i
\(27\) 0 0
\(28\) −1.56942 − 1.14025i −0.296593 − 0.215487i
\(29\) 5.15218 + 5.72208i 0.956737 + 1.06256i 0.997987 + 0.0634149i \(0.0201991\pi\)
−0.0412506 + 0.999149i \(0.513134\pi\)
\(30\) 0 0
\(31\) 5.96439 + 2.65552i 1.07124 + 0.476945i 0.865111 − 0.501581i \(-0.167248\pi\)
0.206125 + 0.978526i \(0.433915\pi\)
\(32\) 2.72743 + 4.72404i 0.482146 + 0.835101i
\(33\) 0 0
\(34\) 0.0907715 − 0.157221i 0.0155672 − 0.0269632i
\(35\) −2.83004 + 2.05614i −0.478363 + 0.347551i
\(36\) 0 0
\(37\) −1.92922 + 5.93753i −0.317162 + 0.976124i 0.657694 + 0.753286i \(0.271532\pi\)
−0.974855 + 0.222838i \(0.928468\pi\)
\(38\) −3.36160 + 1.49668i −0.545324 + 0.242794i
\(39\) 0 0
\(40\) 6.22531 − 1.32323i 0.984308 − 0.209221i
\(41\) −3.88703 + 4.31698i −0.607052 + 0.674199i −0.965816 − 0.259229i \(-0.916532\pi\)
0.358764 + 0.933428i \(0.383198\pi\)
\(42\) 0 0
\(43\) 3.39229 − 5.87562i 0.517320 − 0.896024i −0.482478 − 0.875908i \(-0.660263\pi\)
0.999798 − 0.0201157i \(-0.00640347\pi\)
\(44\) −0.210303 − 5.52194i −0.0317043 − 0.832463i
\(45\) 0 0
\(46\) 0.0399679 − 0.0290384i 0.00589294 − 0.00428147i
\(47\) −0.320574 − 0.0681401i −0.0467605 − 0.00993925i 0.184472 − 0.982838i \(-0.440942\pi\)
−0.231233 + 0.972899i \(0.574276\pi\)
\(48\) 0 0
\(49\) 0.589996 − 5.61344i 0.0842852 − 0.801920i
\(50\) 0.243197 − 2.31387i 0.0343933 − 0.327230i
\(51\) 0 0
\(52\) −6.43117 − 1.36699i −0.891843 − 0.189567i
\(53\) −1.96000 + 1.42402i −0.269227 + 0.195605i −0.714205 − 0.699937i \(-0.753211\pi\)
0.444978 + 0.895541i \(0.353211\pi\)
\(54\) 0 0
\(55\) −9.58718 − 2.71632i −1.29274 − 0.366269i
\(56\) 1.23321 − 2.13598i 0.164795 − 0.285433i
\(57\) 0 0
\(58\) −2.97699 + 3.30629i −0.390899 + 0.434137i
\(59\) −2.24586 + 0.477372i −0.292386 + 0.0621485i −0.351769 − 0.936087i \(-0.614420\pi\)
0.0593835 + 0.998235i \(0.481087\pi\)
\(60\) 0 0
\(61\) −12.5039 + 5.56709i −1.60096 + 0.712792i −0.996480 − 0.0838297i \(-0.973285\pi\)
−0.604478 + 0.796622i \(0.706618\pi\)
\(62\) −1.16575 + 3.58780i −0.148050 + 0.455652i
\(63\) 0 0
\(64\) 0.861321 − 0.625787i 0.107665 − 0.0782233i
\(65\) −5.92799 + 10.2676i −0.735277 + 1.27354i
\(66\) 0 0
\(67\) −5.83989 − 10.1150i −0.713456 − 1.23574i −0.963552 − 0.267521i \(-0.913795\pi\)
0.250096 − 0.968221i \(-0.419538\pi\)
\(68\) −0.478226 − 0.212920i −0.0579934 − 0.0258203i
\(69\) 0 0
\(70\) −1.35248 − 1.50208i −0.161653 − 0.179533i
\(71\) 7.05272 + 5.12410i 0.837004 + 0.608119i 0.921532 − 0.388302i \(-0.126938\pi\)
−0.0845279 + 0.996421i \(0.526938\pi\)
\(72\) 0 0
\(73\) 0.910538 − 2.80235i 0.106570 − 0.327990i −0.883525 − 0.468383i \(-0.844837\pi\)
0.990096 + 0.140393i \(0.0448367\pi\)
\(74\) −3.52850 − 0.750007i −0.410180 − 0.0871865i
\(75\) 0 0
\(76\) 5.30529 + 9.18904i 0.608559 + 1.05405i
\(77\) −3.26835 + 2.05668i −0.372463 + 0.234381i
\(78\) 0 0
\(79\) 0.928965 + 8.83851i 0.104517 + 0.994410i 0.913572 + 0.406677i \(0.133313\pi\)
−0.809055 + 0.587733i \(0.800021\pi\)
\(80\) −1.95735 − 6.02412i −0.218839 − 0.673517i
\(81\) 0 0
\(82\) −2.71551 − 1.97293i −0.299878 − 0.217874i
\(83\) −4.52916 + 2.01651i −0.497140 + 0.221341i −0.639961 − 0.768408i \(-0.721049\pi\)
0.142821 + 0.989749i \(0.454383\pi\)
\(84\) 0 0
\(85\) −0.631634 + 0.701501i −0.0685104 + 0.0760885i
\(86\) 3.58129 + 1.59449i 0.386181 + 0.171939i
\(87\) 0 0
\(88\) 6.95423 − 0.999772i 0.741323 − 0.106576i
\(89\) −2.12862 −0.225634 −0.112817 − 0.993616i \(-0.535987\pi\)
−0.112817 + 0.993616i \(0.535987\pi\)
\(90\) 0 0
\(91\) 1.41981 + 4.36973i 0.148837 + 0.458072i
\(92\) −0.0953208 − 0.105864i −0.00993788 − 0.0110371i
\(93\) 0 0
\(94\) 0.0197945 − 0.188332i 0.00204165 − 0.0194250i
\(95\) 18.7153 − 3.97805i 1.92014 − 0.408139i
\(96\) 0 0
\(97\) 0.00928623 + 0.0883526i 0.000942874 + 0.00897084i 0.994983 − 0.100042i \(-0.0318977\pi\)
−0.994040 + 0.109013i \(0.965231\pi\)
\(98\) 3.26138 0.329449
\(99\) 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 297.2.n.b.289.6 72
3.2 odd 2 99.2.m.b.25.4 yes 72
9.2 odd 6 891.2.f.f.487.4 36
9.4 even 3 inner 297.2.n.b.91.4 72
9.5 odd 6 99.2.m.b.58.6 yes 72
9.7 even 3 891.2.f.e.487.6 36
11.4 even 5 inner 297.2.n.b.235.4 72
33.2 even 10 1089.2.e.o.727.11 36
33.20 odd 10 1089.2.e.p.727.8 36
33.26 odd 10 99.2.m.b.70.6 yes 72
99.2 even 30 9801.2.a.co.1.8 18
99.4 even 15 inner 297.2.n.b.37.6 72
99.20 odd 30 9801.2.a.cm.1.11 18
99.59 odd 30 99.2.m.b.4.4 ✓ 72
99.68 even 30 1089.2.e.o.364.11 36
99.70 even 15 891.2.f.e.730.6 36
99.79 odd 30 9801.2.a.cn.1.11 18
99.86 odd 30 1089.2.e.p.364.8 36
99.92 odd 30 891.2.f.f.730.4 36
99.97 even 15 9801.2.a.cp.1.8 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.m.b.4.4 ✓ 72 99.59 odd 30
99.2.m.b.25.4 yes 72 3.2 odd 2
99.2.m.b.58.6 yes 72 9.5 odd 6
99.2.m.b.70.6 yes 72 33.26 odd 10
297.2.n.b.37.6 72 99.4 even 15 inner
297.2.n.b.91.4 72 9.4 even 3 inner
297.2.n.b.235.4 72 11.4 even 5 inner
297.2.n.b.289.6 72 1.1 even 1 trivial
891.2.f.e.487.6 36 9.7 even 3
891.2.f.e.730.6 36 99.70 even 15
891.2.f.f.487.4 36 9.2 odd 6
891.2.f.f.730.4 36 99.92 odd 30
1089.2.e.o.364.11 36 99.68 even 30
1089.2.e.o.727.11 36 33.2 even 10
1089.2.e.p.364.8 36 99.86 odd 30
1089.2.e.p.727.8 36 33.20 odd 10
9801.2.a.cm.1.11 18 99.20 odd 30
9801.2.a.cn.1.11 18 99.79 odd 30
9801.2.a.co.1.8 18 99.2 even 30
9801.2.a.cp.1.8 18 99.97 even 15