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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 49.8
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.19099 - 0.543906i) q^{3} +(-3.10457 - 2.69013i) q^{5} +(-2.79024 + 1.79318i) q^{7} +(-0.841964 - 0.971678i) q^{9} +(0.218674 + 0.0314405i) q^{11} +(0.704625 - 1.09642i) q^{13} +(2.23433 + 4.89250i) q^{15} +(4.44208 + 1.30431i) q^{17} +(-0.738525 - 2.51518i) q^{19} +(4.29846 - 0.618026i) q^{21} +(0.194426 + 4.79189i) q^{23} +(1.69001 + 11.7543i) q^{25} +(1.58089 + 5.38402i) q^{27} +(-2.29035 + 7.80022i) q^{29} +(-2.87423 - 6.29369i) q^{31} +(-0.243337 - 0.156383i) q^{33} +(13.4864 + 1.93905i) q^{35} +(6.63190 - 5.74657i) q^{37} +(-1.43555 + 0.922571i) q^{39} +(0.436314 - 0.503533i) q^{41} +(-3.09993 - 1.41569i) q^{43} +5.28163i q^{45} +6.77348 q^{47} +(1.66205 - 3.63938i) q^{49} +(-4.58105 - 3.96950i) q^{51} +(1.60743 + 2.50121i) q^{53} +(-0.594310 - 0.685870i) q^{55} +(-0.488450 + 3.39724i) q^{57} +(-5.57839 + 8.68015i) q^{59} +(0.489270 - 0.223442i) q^{61} +(4.09167 + 1.20142i) q^{63} +(-5.13706 + 1.50838i) q^{65} +(-8.31164 + 1.19503i) q^{67} +(2.37478 - 5.81283i) q^{69} +(0.535906 + 3.72731i) q^{71} +(-6.36470 + 1.86884i) q^{73} +(4.38044 - 14.9184i) q^{75} +(-0.666531 + 0.304395i) q^{77} +(8.33276 + 5.35514i) q^{79} +(0.496650 - 3.45428i) q^{81} +(0.921871 - 0.798806i) q^{83} +(-10.2820 - 15.9991i) q^{85} +(6.97037 - 8.04423i) q^{87} +(0.123211 - 0.269794i) q^{89} +4.32279i q^{91} +9.05902i q^{93} +(-4.47336 + 9.79530i) q^{95} +(1.60298 - 1.84994i) q^{97} +(-0.153565 - 0.238952i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) 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
\(3\) −1.19099 − 0.543906i −0.687617 − 0.314024i 0.0407910 − 0.999168i \(-0.487012\pi\)
−0.728408 + 0.685143i \(0.759740\pi\)
\(4\) 0 0
\(5\) −3.10457 − 2.69013i −1.38841 − 1.20306i −0.953053 − 0.302802i \(-0.902078\pi\)
−0.435354 − 0.900260i \(-0.643377\pi\)
\(6\) 0 0
\(7\) −2.79024 + 1.79318i −1.05461 + 0.677758i −0.948558 − 0.316602i \(-0.897458\pi\)
−0.106054 + 0.994360i \(0.533822\pi\)
\(8\) 0 0
\(9\) −0.841964 − 0.971678i −0.280655 − 0.323893i
\(10\) 0 0
\(11\) 0.218674 + 0.0314405i 0.0659326 + 0.00947968i 0.175202 − 0.984532i \(-0.443942\pi\)
−0.109270 + 0.994012i \(0.534851\pi\)
\(12\) 0 0
\(13\) 0.704625 − 1.09642i 0.195428 − 0.304092i −0.729685 − 0.683783i \(-0.760333\pi\)
0.925113 + 0.379691i \(0.123970\pi\)
\(14\) 0 0
\(15\) 2.23433 + 4.89250i 0.576902 + 1.26324i
\(16\) 0 0
\(17\) 4.44208 + 1.30431i 1.07736 + 0.316343i 0.771824 − 0.635837i \(-0.219345\pi\)
0.305540 + 0.952179i \(0.401163\pi\)
\(18\) 0 0
\(19\) −0.738525 − 2.51518i −0.169429 − 0.577023i −0.999804 − 0.0198170i \(-0.993692\pi\)
0.830374 − 0.557206i \(-0.188127\pi\)
\(20\) 0 0
\(21\) 4.29846 − 0.618026i 0.938002 − 0.134864i
\(22\) 0 0
\(23\) 0.194426 + 4.79189i 0.0405406 + 0.999178i
\(24\) 0 0
\(25\) 1.69001 + 11.7543i 0.338002 + 2.35086i
\(26\) 0 0
\(27\) 1.58089 + 5.38402i 0.304243 + 1.03616i
\(28\) 0 0
\(29\) −2.29035 + 7.80022i −0.425307 + 1.44846i 0.416723 + 0.909034i \(0.363179\pi\)
−0.842030 + 0.539431i \(0.818640\pi\)
\(30\) 0 0
\(31\) −2.87423 − 6.29369i −0.516227 − 1.13038i −0.970848 − 0.239695i \(-0.922953\pi\)
0.454621 − 0.890685i \(-0.349775\pi\)
\(32\) 0 0
\(33\) −0.243337 − 0.156383i −0.0423596 − 0.0272228i
\(34\) 0 0
\(35\) 13.4864 + 1.93905i 2.27962 + 0.327759i
\(36\) 0 0
\(37\) 6.63190 − 5.74657i 1.09028 − 0.944731i 0.0915822 − 0.995798i \(-0.470808\pi\)
0.998695 + 0.0510667i \(0.0162621\pi\)
\(38\) 0 0
\(39\) −1.43555 + 0.922571i −0.229872 + 0.147730i
\(40\) 0 0
\(41\) 0.436314 − 0.503533i 0.0681408 − 0.0786387i −0.720656 − 0.693293i \(-0.756159\pi\)
0.788796 + 0.614655i \(0.210705\pi\)
\(42\) 0 0
\(43\) −3.09993 − 1.41569i −0.472736 − 0.215891i 0.164777 − 0.986331i \(-0.447309\pi\)
−0.637513 + 0.770440i \(0.720037\pi\)
\(44\) 0 0
\(45\) 5.28163i 0.787339i
\(46\) 0 0
\(47\) 6.77348 0.988013 0.494007 − 0.869458i \(-0.335532\pi\)
0.494007 + 0.869458i \(0.335532\pi\)
\(48\) 0 0
\(49\) 1.66205 − 3.63938i 0.237436 − 0.519912i
\(50\) 0 0
\(51\) −4.58105 − 3.96950i −0.641475 − 0.555841i
\(52\) 0 0
\(53\) 1.60743 + 2.50121i 0.220798 + 0.343568i 0.933926 − 0.357465i \(-0.116359\pi\)
−0.713129 + 0.701033i \(0.752723\pi\)
\(54\) 0 0
\(55\) −0.594310 − 0.685870i −0.0801367 − 0.0924827i
\(56\) 0 0
\(57\) −0.488450 + 3.39724i −0.0646967 + 0.449976i
\(58\) 0 0
\(59\) −5.57839 + 8.68015i −0.726245 + 1.13006i 0.260134 + 0.965572i \(0.416233\pi\)
−0.986379 + 0.164487i \(0.947403\pi\)
\(60\) 0 0
\(61\) 0.489270 − 0.223442i 0.0626446 − 0.0286088i −0.383846 − 0.923397i \(-0.625401\pi\)
0.446491 + 0.894788i \(0.352673\pi\)
\(62\) 0 0
\(63\) 4.09167 + 1.20142i 0.515503 + 0.151365i
\(64\) 0 0
\(65\) −5.13706 + 1.50838i −0.637174 + 0.187091i
\(66\) 0 0
\(67\) −8.31164 + 1.19503i −1.01543 + 0.145997i −0.629881 − 0.776691i \(-0.716896\pi\)
−0.385547 + 0.922688i \(0.625987\pi\)
\(68\) 0 0
\(69\) 2.37478 − 5.81283i 0.285890 − 0.699783i
\(70\) 0 0
\(71\) 0.535906 + 3.72731i 0.0636003 + 0.442350i 0.996595 + 0.0824585i \(0.0262772\pi\)
−0.932994 + 0.359891i \(0.882814\pi\)
\(72\) 0 0
\(73\) −6.36470 + 1.86884i −0.744931 + 0.218732i −0.632105 − 0.774883i \(-0.717809\pi\)
−0.112827 + 0.993615i \(0.535991\pi\)
\(74\) 0 0
\(75\) 4.38044 − 14.9184i 0.505810 − 1.72263i
\(76\) 0 0
\(77\) −0.666531 + 0.304395i −0.0759583 + 0.0346890i
\(78\) 0 0
\(79\) 8.33276 + 5.35514i 0.937509 + 0.602500i 0.917687 − 0.397303i \(-0.130054\pi\)
0.0198214 + 0.999804i \(0.493690\pi\)
\(80\) 0 0
\(81\) 0.496650 − 3.45428i 0.0551833 − 0.383808i
\(82\) 0 0
\(83\) 0.921871 − 0.798806i 0.101188 − 0.0876803i −0.602797 − 0.797894i \(-0.705947\pi\)
0.703986 + 0.710214i \(0.251402\pi\)
\(84\) 0 0
\(85\) −10.2820 − 15.9991i −1.11524 − 1.73535i
\(86\) 0 0
\(87\) 6.97037 − 8.04423i 0.747302 − 0.862432i
\(88\) 0 0
\(89\) 0.123211 − 0.269794i 0.0130603 − 0.0285981i −0.902989 − 0.429663i \(-0.858632\pi\)
0.916050 + 0.401065i \(0.131360\pi\)
\(90\) 0 0
\(91\) 4.32279i 0.453151i
\(92\) 0 0
\(93\) 9.05902i 0.939377i
\(94\) 0 0
\(95\) −4.47336 + 9.79530i −0.458957 + 1.00498i
\(96\) 0 0
\(97\) 1.60298 − 1.84994i 0.162758 − 0.187833i −0.668512 − 0.743701i \(-0.733069\pi\)
0.831271 + 0.555868i \(0.187614\pi\)
\(98\) 0 0
\(99\) −0.153565 − 0.238952i −0.0154339 − 0.0240156i
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.49.8 220
4.3 odd 2 184.2.p.a.141.12 yes 220
8.3 odd 2 184.2.p.a.141.9 yes 220
8.5 even 2 inner 736.2.x.a.49.15 220
23.8 even 11 inner 736.2.x.a.721.15 220
92.31 odd 22 184.2.p.a.77.9 ✓ 220
184.77 even 22 inner 736.2.x.a.721.8 220
184.123 odd 22 184.2.p.a.77.12 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.9 ✓ 220 92.31 odd 22
184.2.p.a.77.12 yes 220 184.123 odd 22
184.2.p.a.141.9 yes 220 8.3 odd 2
184.2.p.a.141.12 yes 220 4.3 odd 2
736.2.x.a.49.8 220 1.1 even 1 trivial
736.2.x.a.49.15 220 8.5 even 2 inner
736.2.x.a.721.8 220 184.77 even 22 inner
736.2.x.a.721.15 220 23.8 even 11 inner