↑

Properties

Label 912.2.cc.c.641.1
Level $912$
Weight $2$
Character 912.641
Analytic conductor $7.282$
Analytic rank $0$
Dimension $18$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [912,2,Mod(257,912)] 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("912.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 9, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.cc (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,-3,0,0,0,0,0,-3,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 641.1
Root \(1.69944 + 0.334495i\) of defining polynomial
Character \(\chi\) \(=\) 912.641
Dual form 912.2.cc.c.737.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.08684 + 1.34862i) q^{3} +(0.343148 - 0.408948i) q^{5} +(0.716507 + 1.24103i) q^{7} +(-0.637553 - 2.93147i) q^{9} +(-1.25645 - 0.725411i) q^{11} +(2.94737 - 0.519701i) q^{13} +(0.178568 + 0.907238i) q^{15} +(1.89590 + 5.20894i) q^{17} +(4.35653 - 0.143752i) q^{19} +(-2.45240 - 0.382503i) q^{21} +(-0.396438 - 0.472456i) q^{23} +(0.818753 + 4.64338i) q^{25} +(4.64636 + 2.32623i) q^{27} +(-4.97822 - 1.81193i) q^{29} +(4.28601 - 2.47453i) q^{31} +(2.34386 - 0.906066i) q^{33} +(0.753384 + 0.132842i) q^{35} +6.41883i q^{37} +(-2.50245 + 4.53972i) q^{39} +(-1.37347 + 7.78933i) q^{41} +(-4.88757 - 4.10116i) q^{43} +(-1.41759 - 0.745203i) q^{45} +(-4.37381 + 12.0169i) q^{47} +(2.47323 - 4.28377i) q^{49} +(-9.08542 - 3.10444i) q^{51} +(-1.41439 + 1.18682i) q^{53} +(-0.727804 + 0.264899i) q^{55} +(-4.54099 + 6.03154i) q^{57} +(1.75650 - 0.639313i) q^{59} +(9.02625 - 7.57392i) q^{61} +(3.18122 - 2.89164i) q^{63} +(0.798855 - 1.38366i) q^{65} +(-3.17216 + 8.71543i) q^{67} +(1.06803 - 0.0211592i) q^{69} +(9.59384 + 8.05019i) q^{71} +(-2.80621 + 15.9148i) q^{73} +(-7.15201 - 3.94243i) q^{75} -2.07905i q^{77} +(7.87896 + 1.38927i) q^{79} +(-8.18705 + 3.73794i) q^{81} +(-4.29627 + 2.48045i) q^{83} +(2.78076 + 1.01211i) q^{85} +(7.85414 - 4.74446i) q^{87} +(-0.832120 - 4.71919i) q^{89} +(2.75678 + 3.28540i) q^{91} +(-1.32101 + 8.46962i) q^{93} +(1.43615 - 1.83092i) q^{95} +(-2.83601 - 7.79188i) q^{97} +(-1.32547 + 4.14573i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{3} - 3 q^{9} - 12 q^{13} - 18 q^{15} + 6 q^{17} + 6 q^{19} - 18 q^{25} + 6 q^{27} - 6 q^{29} - 24 q^{33} + 24 q^{35} - 6 q^{39} + 3 q^{41} + 6 q^{43} - 54 q^{45} - 30 q^{47} + 21 q^{49} - 42 q^{51}+ \cdots + 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(1\) \(-1\) \(1\)

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
\(3\) −1.08684 + 1.34862i −0.627488 + 0.778626i
\(4\) 0 0
\(5\) 0.343148 − 0.408948i 0.153461 − 0.182887i −0.683837 − 0.729635i \(-0.739690\pi\)
0.837297 + 0.546748i \(0.184134\pi\)
\(6\) 0 0
\(7\) 0.716507 + 1.24103i 0.270814 + 0.469064i 0.969071 − 0.246784i \(-0.0793737\pi\)
−0.698256 + 0.715848i \(0.746040\pi\)
\(8\) 0 0
\(9\) −0.637553 − 2.93147i −0.212518 − 0.977157i
\(10\) 0 0
\(11\) −1.25645 − 0.725411i −0.378834 − 0.218720i 0.298477 − 0.954417i \(-0.403521\pi\)
−0.677311 + 0.735697i \(0.736855\pi\)
\(12\) 0 0
\(13\) 2.94737 − 0.519701i 0.817454 − 0.144139i 0.250741 − 0.968054i \(-0.419326\pi\)
0.566713 + 0.823915i \(0.308215\pi\)
\(14\) 0 0
\(15\) 0.178568 + 0.907238i 0.0461061 + 0.234248i
\(16\) 0 0
\(17\) 1.89590 + 5.20894i 0.459823 + 1.26335i 0.925618 + 0.378460i \(0.123546\pi\)
−0.465795 + 0.884893i \(0.654232\pi\)
\(18\) 0 0
\(19\) 4.35653 − 0.143752i 0.999456 − 0.0329790i
\(20\) 0 0
\(21\) −2.45240 − 0.382503i −0.535158 − 0.0834690i
\(22\) 0 0
\(23\) −0.396438 − 0.472456i −0.0826630 − 0.0985139i 0.723128 − 0.690714i \(-0.242704\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(24\) 0 0
\(25\) 0.818753 + 4.64338i 0.163751 + 0.928676i
\(26\) 0 0
\(27\) 4.64636 + 2.32623i 0.894192 + 0.447683i
\(28\) 0 0
\(29\) −4.97822 − 1.81193i −0.924433 − 0.336466i −0.164432 − 0.986388i \(-0.552579\pi\)
−0.760001 + 0.649922i \(0.774801\pi\)
\(30\) 0 0
\(31\) 4.28601 − 2.47453i 0.769790 − 0.444439i −0.0630096 − 0.998013i \(-0.520070\pi\)
0.832800 + 0.553574i \(0.186737\pi\)
\(32\) 0 0
\(33\) 2.34386 − 0.906066i 0.408014 − 0.157726i
\(34\) 0 0
\(35\) 0.753384 + 0.132842i 0.127345 + 0.0224544i
\(36\) 0 0
\(37\) 6.41883i 1.05525i 0.849478 + 0.527625i \(0.176917\pi\)
−0.849478 + 0.527625i \(0.823083\pi\)
\(38\) 0 0
\(39\) −2.50245 + 4.53972i −0.400712 + 0.726937i
\(40\) 0 0
\(41\) −1.37347 + 7.78933i −0.214500 + 1.21649i 0.667273 + 0.744814i \(0.267462\pi\)
−0.881772 + 0.471675i \(0.843650\pi\)
\(42\) 0 0
\(43\) −4.88757 − 4.10116i −0.745348 − 0.625421i 0.188920 − 0.981992i \(-0.439501\pi\)
−0.934268 + 0.356571i \(0.883946\pi\)
\(44\) 0 0
\(45\) −1.41759 − 0.745203i −0.211323 − 0.111088i
\(46\) 0 0
\(47\) −4.37381 + 12.0169i −0.637985 + 1.75285i 0.0199780 + 0.999800i \(0.493640\pi\)
−0.657963 + 0.753050i \(0.728582\pi\)
\(48\) 0 0
\(49\) 2.47323 − 4.28377i 0.353319 − 0.611967i
\(50\) 0 0
\(51\) −9.08542 − 3.10444i −1.27221 − 0.434709i
\(52\) 0 0
\(53\) −1.41439 + 1.18682i −0.194282 + 0.163022i −0.734739 − 0.678349i \(-0.762696\pi\)
0.540458 + 0.841371i \(0.318251\pi\)
\(54\) 0 0
\(55\) −0.727804 + 0.264899i −0.0981370 + 0.0357190i
\(56\) 0 0
\(57\) −4.54099 + 6.03154i −0.601468 + 0.798897i
\(58\) 0 0
\(59\) 1.75650 − 0.639313i 0.228676 − 0.0832314i −0.225141 − 0.974326i \(-0.572284\pi\)
0.453817 + 0.891095i \(0.350062\pi\)
\(60\) 0 0
\(61\) 9.02625 − 7.57392i 1.15569 − 0.969742i 0.155856 − 0.987780i \(-0.450186\pi\)
0.999837 + 0.0180382i \(0.00574206\pi\)
\(62\) 0 0
\(63\) 3.18122 − 2.89164i 0.400797 − 0.364313i
\(64\) 0 0
\(65\) 0.798855 − 1.38366i 0.0990857 − 0.171622i
\(66\) 0 0
\(67\) −3.17216 + 8.71543i −0.387541 + 1.06476i 0.580564 + 0.814215i \(0.302832\pi\)
−0.968105 + 0.250545i \(0.919390\pi\)
\(68\) 0 0
\(69\) 1.06803 − 0.0211592i 0.128576 − 0.00254727i
\(70\) 0 0
\(71\) 9.59384 + 8.05019i 1.13858 + 0.955382i 0.999391 − 0.0348843i \(-0.0111063\pi\)
0.139188 + 0.990266i \(0.455551\pi\)
\(72\) 0 0
\(73\) −2.80621 + 15.9148i −0.328442 + 1.86269i 0.155852 + 0.987780i \(0.450188\pi\)
−0.484294 + 0.874905i \(0.660923\pi\)
\(74\) 0 0
\(75\) −7.15201 − 3.94243i −0.825843 − 0.455232i
\(76\) 0 0
\(77\) − 2.07905i − 0.236930i
\(78\) 0 0
\(79\) 7.87896 + 1.38927i 0.886452 + 0.156305i 0.598293 − 0.801277i \(-0.295846\pi\)
0.288159 + 0.957583i \(0.406957\pi\)
\(80\) 0 0
\(81\) −8.18705 + 3.73794i −0.909673 + 0.415326i
\(82\) 0 0
\(83\) −4.29627 + 2.48045i −0.471577 + 0.272265i −0.716900 − 0.697176i \(-0.754439\pi\)
0.245323 + 0.969442i \(0.421106\pi\)
\(84\) 0 0
\(85\) 2.78076 + 1.01211i 0.301616 + 0.109779i
\(86\) 0 0
\(87\) 7.85414 − 4.74446i 0.842052 − 0.508659i
\(88\) 0 0
\(89\) −0.832120 − 4.71919i −0.0882046 − 0.500233i −0.996619 − 0.0821621i \(-0.973817\pi\)
0.908414 − 0.418071i \(-0.137294\pi\)
\(90\) 0 0
\(91\) 2.75678 + 3.28540i 0.288989 + 0.344403i
\(92\) 0 0
\(93\) −1.32101 + 8.46962i −0.136983 + 0.878259i
\(94\) 0 0
\(95\) 1.43615 − 1.83092i 0.147346 − 0.187849i
\(96\) 0 0
\(97\) −2.83601 − 7.79188i −0.287954 − 0.791146i −0.996352 − 0.0853335i \(-0.972804\pi\)
0.708399 − 0.705812i \(-0.249418\pi\)
\(98\) 0 0
\(99\) −1.32547 + 4.14573i −0.133215 + 0.416662i
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 912.2.cc.c.641.1 18
3.2 odd 2 912.2.cc.d.641.2 18
4.3 odd 2 114.2.l.b.71.3 yes 18
12.11 even 2 114.2.l.a.71.2 yes 18
19.15 odd 18 912.2.cc.d.737.2 18
57.53 even 18 inner 912.2.cc.c.737.1 18
76.15 even 18 114.2.l.a.53.2 ✓ 18
228.167 odd 18 114.2.l.b.53.3 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.l.a.53.2 ✓ 18 76.15 even 18
114.2.l.a.71.2 yes 18 12.11 even 2
114.2.l.b.53.3 yes 18 228.167 odd 18
114.2.l.b.71.3 yes 18 4.3 odd 2
912.2.cc.c.641.1 18 1.1 even 1 trivial
912.2.cc.c.737.1 18 57.53 even 18 inner
912.2.cc.d.641.2 18 3.2 odd 2
912.2.cc.d.737.2 18 19.15 odd 18