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Properties

Label 368.2.m.e.177.2
Level $368$
Weight $2$
Character 368.177
Analytic conductor $2.938$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [368,2,Mod(49,368)] 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("368.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(368, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 0, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.m (of order \(11\), degree \(10\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.93849479438\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 177.2
Character \(\chi\) \(=\) 368.177
Dual form 368.2.m.e.289.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.21882 + 0.783290i) q^{3} +(-1.24491 - 2.72597i) q^{5} +(-1.62369 + 0.476758i) q^{7} +(-0.374258 + 0.819511i) q^{9} +(3.72943 + 4.30399i) q^{11} +(6.20950 + 1.82327i) q^{13} +(3.65254 + 2.34735i) q^{15} +(-0.846356 + 5.88654i) q^{17} +(0.0398636 + 0.277257i) q^{19} +(1.60555 - 1.85290i) q^{21} +(-4.79340 - 0.152709i) q^{23} +(-2.60679 + 3.00840i) q^{25} +(-0.804325 - 5.59421i) q^{27} +(-0.171530 + 1.19302i) q^{29} +(2.51819 + 1.61834i) q^{31} +(-7.91679 - 2.32458i) q^{33} +(3.32097 + 3.83260i) q^{35} +(0.110238 - 0.241387i) q^{37} +(-8.99644 + 2.64159i) q^{39} +(2.41581 + 5.28989i) q^{41} +(4.76425 - 3.06179i) q^{43} +2.69988 q^{45} +1.51532 q^{47} +(-3.47971 + 2.23627i) q^{49} +(-3.57931 - 7.83759i) q^{51} +(-6.33809 + 1.86103i) q^{53} +(7.08974 - 15.5244i) q^{55} +(-0.265760 - 0.306703i) q^{57} +(3.08055 + 0.904532i) q^{59} +(7.37216 + 4.73780i) q^{61} +(0.216970 - 1.50906i) q^{63} +(-2.76007 - 19.1967i) q^{65} +(-2.04000 + 2.35428i) q^{67} +(5.96192 - 3.56850i) q^{69} +(-1.43689 + 1.65826i) q^{71} +(-1.80221 - 12.5346i) q^{73} +(0.820770 - 5.70858i) q^{75} +(-8.10739 - 5.21030i) q^{77} +(0.627270 + 0.184183i) q^{79} +(3.59227 + 4.14571i) q^{81} +(-5.74370 + 12.5769i) q^{83} +(17.1001 - 5.02105i) q^{85} +(-0.725415 - 1.58844i) q^{87} +(-8.24726 + 5.30019i) q^{89} -10.9516 q^{91} -4.33686 q^{93} +(0.706168 - 0.453826i) q^{95} +(1.58933 + 3.48015i) q^{97} +(-4.92294 + 1.44550i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 2 q^{3} + 13 q^{7} + 21 q^{9} - 2 q^{11} + 2 q^{15} - 22 q^{17} - 3 q^{19} + 2 q^{21} - q^{23} + 13 q^{25} + 31 q^{27} + 7 q^{29} - 18 q^{31} - 8 q^{33} - 41 q^{35} - 62 q^{37} - 6 q^{39} - 15 q^{41}+ \cdots + 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\) \(277\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{11}\right)\) \(1\)

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
\(3\) −1.21882 + 0.783290i −0.703688 + 0.452233i −0.842928 − 0.538026i \(-0.819170\pi\)
0.139240 + 0.990259i \(0.455534\pi\)
\(4\) 0 0
\(5\) −1.24491 − 2.72597i −0.556739 − 1.21909i −0.953563 − 0.301194i \(-0.902615\pi\)
0.396824 − 0.917895i \(-0.370112\pi\)
\(6\) 0 0
\(7\) −1.62369 + 0.476758i −0.613697 + 0.180198i −0.573785 − 0.819006i \(-0.694526\pi\)
−0.0399111 + 0.999203i \(0.512707\pi\)
\(8\) 0 0
\(9\) −0.374258 + 0.819511i −0.124753 + 0.273170i
\(10\) 0 0
\(11\) 3.72943 + 4.30399i 1.12447 + 1.29770i 0.949724 + 0.313087i \(0.101363\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(12\) 0 0
\(13\) 6.20950 + 1.82327i 1.72221 + 0.505685i 0.985376 − 0.170393i \(-0.0545038\pi\)
0.736830 + 0.676078i \(0.236322\pi\)
\(14\) 0 0
\(15\) 3.65254 + 2.34735i 0.943083 + 0.606083i
\(16\) 0 0
\(17\) −0.846356 + 5.88654i −0.205272 + 1.42769i 0.583052 + 0.812435i \(0.301858\pi\)
−0.788324 + 0.615260i \(0.789051\pi\)
\(18\) 0 0
\(19\) 0.0398636 + 0.277257i 0.00914533 + 0.0636072i 0.993883 − 0.110438i \(-0.0352253\pi\)
−0.984738 + 0.174045i \(0.944316\pi\)
\(20\) 0 0
\(21\) 1.60555 − 1.85290i 0.350360 − 0.404337i
\(22\) 0 0
\(23\) −4.79340 − 0.152709i −0.999493 − 0.0318421i
\(24\) 0 0
\(25\) −2.60679 + 3.00840i −0.521359 + 0.601680i
\(26\) 0 0
\(27\) −0.804325 − 5.59421i −0.154792 − 1.07661i
\(28\) 0 0
\(29\) −0.171530 + 1.19302i −0.0318524 + 0.221538i −0.999530 − 0.0306519i \(-0.990242\pi\)
0.967678 + 0.252190i \(0.0811508\pi\)
\(30\) 0 0
\(31\) 2.51819 + 1.61834i 0.452280 + 0.290663i 0.746881 − 0.664958i \(-0.231550\pi\)
−0.294601 + 0.955620i \(0.595187\pi\)
\(32\) 0 0
\(33\) −7.91679 − 2.32458i −1.37814 − 0.404657i
\(34\) 0 0
\(35\) 3.32097 + 3.83260i 0.561346 + 0.647828i
\(36\) 0 0
\(37\) 0.110238 − 0.241387i 0.0181230 − 0.0396838i −0.900354 − 0.435158i \(-0.856692\pi\)
0.918477 + 0.395474i \(0.129420\pi\)
\(38\) 0 0
\(39\) −8.99644 + 2.64159i −1.44058 + 0.422993i
\(40\) 0 0
\(41\) 2.41581 + 5.28989i 0.377286 + 0.826142i 0.999077 + 0.0429584i \(0.0136783\pi\)
−0.621790 + 0.783184i \(0.713594\pi\)
\(42\) 0 0
\(43\) 4.76425 − 3.06179i 0.726541 − 0.466919i −0.124366 − 0.992236i \(-0.539690\pi\)
0.850907 + 0.525317i \(0.176053\pi\)
\(44\) 0 0
\(45\) 2.69988 0.402474
\(46\) 0 0
\(47\) 1.51532 0.221032 0.110516 − 0.993874i \(-0.464750\pi\)
0.110516 + 0.993874i \(0.464750\pi\)
\(48\) 0 0
\(49\) −3.47971 + 2.23627i −0.497101 + 0.319468i
\(50\) 0 0
\(51\) −3.57931 − 7.83759i −0.501203 − 1.09748i
\(52\) 0 0
\(53\) −6.33809 + 1.86103i −0.870604 + 0.255632i −0.686372 − 0.727251i \(-0.740798\pi\)
−0.184232 + 0.982883i \(0.558980\pi\)
\(54\) 0 0
\(55\) 7.08974 − 15.5244i 0.955980 − 2.09330i
\(56\) 0 0
\(57\) −0.265760 − 0.306703i −0.0352007 − 0.0406238i
\(58\) 0 0
\(59\) 3.08055 + 0.904532i 0.401054 + 0.117760i 0.476038 − 0.879425i \(-0.342072\pi\)
−0.0749841 + 0.997185i \(0.523891\pi\)
\(60\) 0 0
\(61\) 7.37216 + 4.73780i 0.943908 + 0.606613i 0.919500 − 0.393089i \(-0.128594\pi\)
0.0244076 + 0.999702i \(0.492230\pi\)
\(62\) 0 0
\(63\) 0.216970 − 1.50906i 0.0273357 − 0.190124i
\(64\) 0 0
\(65\) −2.76007 − 19.1967i −0.342344 − 2.38106i
\(66\) 0 0
\(67\) −2.04000 + 2.35428i −0.249225 + 0.287621i −0.866553 − 0.499085i \(-0.833670\pi\)
0.617328 + 0.786706i \(0.288215\pi\)
\(68\) 0 0
\(69\) 5.96192 − 3.56850i 0.717731 − 0.429597i
\(70\) 0 0
\(71\) −1.43689 + 1.65826i −0.170528 + 0.196800i −0.834580 − 0.550886i \(-0.814290\pi\)
0.664052 + 0.747686i \(0.268835\pi\)
\(72\) 0 0
\(73\) −1.80221 − 12.5346i −0.210933 − 1.46707i −0.770051 − 0.637982i \(-0.779769\pi\)
0.559118 − 0.829088i \(-0.311140\pi\)
\(74\) 0 0
\(75\) 0.820770 − 5.70858i 0.0947744 − 0.659170i
\(76\) 0 0
\(77\) −8.10739 − 5.21030i −0.923923 − 0.593769i
\(78\) 0 0
\(79\) 0.627270 + 0.184183i 0.0705734 + 0.0207222i 0.316829 − 0.948483i \(-0.397382\pi\)
−0.246255 + 0.969205i \(0.579200\pi\)
\(80\) 0 0
\(81\) 3.59227 + 4.14571i 0.399142 + 0.460634i
\(82\) 0 0
\(83\) −5.74370 + 12.5769i −0.630453 + 1.38050i 0.277215 + 0.960808i \(0.410589\pi\)
−0.907667 + 0.419691i \(0.862139\pi\)
\(84\) 0 0
\(85\) 17.1001 − 5.02105i 1.85477 − 0.544610i
\(86\) 0 0
\(87\) −0.725415 − 1.58844i −0.0777727 − 0.170298i
\(88\) 0 0
\(89\) −8.24726 + 5.30019i −0.874208 + 0.561819i −0.899037 − 0.437873i \(-0.855732\pi\)
0.0248295 + 0.999692i \(0.492096\pi\)
\(90\) 0 0
\(91\) −10.9516 −1.14804
\(92\) 0 0
\(93\) −4.33686 −0.449711
\(94\) 0 0
\(95\) 0.706168 − 0.453826i 0.0724513 − 0.0465616i
\(96\) 0 0
\(97\) 1.58933 + 3.48015i 0.161372 + 0.353355i 0.972995 − 0.230826i \(-0.0741429\pi\)
−0.811623 + 0.584182i \(0.801416\pi\)
\(98\) 0 0
\(99\) −4.92294 + 1.44550i −0.494774 + 0.145279i
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 368.2.m.e.177.2 30
4.3 odd 2 184.2.i.b.177.2 yes 30
23.6 even 11 8464.2.a.cg.1.5 15
23.13 even 11 inner 368.2.m.e.289.2 30
23.17 odd 22 8464.2.a.ch.1.5 15
92.59 odd 22 184.2.i.b.105.2 ✓ 30
92.63 even 22 4232.2.a.ba.1.11 15
92.75 odd 22 4232.2.a.bb.1.11 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.105.2 ✓ 30 92.59 odd 22
184.2.i.b.177.2 yes 30 4.3 odd 2
368.2.m.e.177.2 30 1.1 even 1 trivial
368.2.m.e.289.2 30 23.13 even 11 inner
4232.2.a.ba.1.11 15 92.63 even 22
4232.2.a.bb.1.11 15 92.75 odd 22
8464.2.a.cg.1.5 15 23.6 even 11
8464.2.a.ch.1.5 15 23.17 odd 22