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Properties

Label 324.2.l.a.179.1
Level $324$
Weight $2$
Character 324.179
Analytic conductor $2.587$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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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] = [324,2,Mod(35,324)] 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("324.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.l (of order \(18\), degree \(6\), 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: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 179.1
Character \(\chi\) \(=\) 324.179
Dual form 324.2.l.a.143.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.36831 + 0.357407i) q^{2} +(1.74452 - 0.978083i) q^{4} +(-0.532084 - 0.0938207i) q^{5} +(-1.18790 - 1.41568i) q^{7} +(-2.03746 + 1.96182i) q^{8} +(0.761586 - 0.0617949i) q^{10} +(-0.867520 - 4.91995i) q^{11} +(-1.52553 + 0.555249i) q^{13} +(2.13138 + 1.51252i) q^{14} +(2.08671 - 3.41257i) q^{16} +(0.683630 + 0.394694i) q^{17} +(-2.86200 + 1.65238i) q^{19} +(-1.02000 + 0.356750i) q^{20} +(2.94546 + 6.42194i) q^{22} +(-3.15234 - 2.64512i) q^{23} +(-4.42415 - 1.61026i) q^{25} +(1.88895 - 1.30499i) q^{26} +(-3.45697 - 1.30783i) q^{28} +(3.24174 - 8.90660i) q^{29} +(3.70462 - 4.41499i) q^{31} +(-1.63558 + 5.41525i) q^{32} +(-1.07648 - 0.295728i) q^{34} +(0.499242 + 0.864712i) q^{35} +(5.53013 - 9.57846i) q^{37} +(3.32552 - 3.28386i) q^{38} +(1.26816 - 0.852696i) q^{40} +(2.62821 + 7.22095i) q^{41} +(-7.88044 + 1.38953i) q^{43} +(-6.32553 - 7.73445i) q^{44} +(5.25874 + 2.49267i) q^{46} +(-2.39693 + 2.01126i) q^{47} +(0.622483 - 3.53027i) q^{49} +(6.62911 + 0.622106i) q^{50} +(-2.11825 + 2.46074i) q^{52} +0.933485i q^{53} +2.69922i q^{55} +(5.19762 + 0.553960i) q^{56} +(-1.25241 + 13.3456i) q^{58} +(0.461077 - 2.61490i) q^{59} +(-4.29949 + 3.60770i) q^{61} +(-3.49110 + 7.36511i) q^{62} +(0.302524 - 7.99428i) q^{64} +(0.863806 - 0.152312i) q^{65} +(2.15593 + 5.92336i) q^{67} +(1.57865 + 0.0199049i) q^{68} +(-0.992169 - 1.00476i) q^{70} +(-4.53203 + 7.84970i) q^{71} +(5.77108 + 9.99580i) q^{73} +(-4.14350 + 15.0828i) q^{74} +(-3.37666 + 5.68189i) q^{76} +(-5.93457 + 7.07254i) q^{77} +(-0.275423 + 0.756719i) q^{79} +(-1.43047 + 1.62000i) q^{80} +(-6.17702 - 8.94113i) q^{82} +(15.0247 + 5.46855i) q^{83} +(-0.326718 - 0.274149i) q^{85} +(10.2862 - 4.71783i) q^{86} +(11.4196 + 8.32231i) q^{88} +(5.41508 - 3.12640i) q^{89} +(2.59824 + 1.50009i) q^{91} +(-8.08647 - 1.53123i) q^{92} +(2.56089 - 3.60870i) q^{94} +(1.67785 - 0.610689i) q^{95} +(1.63933 + 9.29708i) q^{97} +(0.409997 + 5.05297i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 6 q^{2} - 6 q^{4} + 12 q^{5} + 9 q^{8} - 3 q^{10} - 12 q^{13} + 21 q^{14} - 6 q^{16} + 18 q^{17} + 27 q^{20} - 6 q^{22} - 12 q^{25} - 12 q^{28} + 24 q^{29} - 24 q^{32} - 12 q^{34} - 6 q^{37} - 18 q^{38}+ \cdots + 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36831 + 0.357407i −0.967538 + 0.252725i
\(3\) 0 0
\(4\) 1.74452 − 0.978083i 0.872260 − 0.489042i
\(5\) −0.532084 − 0.0938207i −0.237955 − 0.0419579i 0.0533987 − 0.998573i \(-0.482995\pi\)
−0.291354 + 0.956615i \(0.594106\pi\)
\(6\) 0 0
\(7\) −1.18790 − 1.41568i −0.448984 − 0.535078i 0.493315 − 0.869851i \(-0.335785\pi\)
−0.942299 + 0.334773i \(0.891341\pi\)
\(8\) −2.03746 + 1.96182i −0.720352 + 0.693608i
\(9\) 0 0
\(10\) 0.761586 − 0.0617949i 0.240835 − 0.0195413i
\(11\) −0.867520 − 4.91995i −0.261567 − 1.48342i −0.778635 − 0.627477i \(-0.784088\pi\)
0.517068 − 0.855944i \(-0.327023\pi\)
\(12\) 0 0
\(13\) −1.52553 + 0.555249i −0.423107 + 0.153998i −0.544793 − 0.838570i \(-0.683392\pi\)
0.121687 + 0.992569i \(0.461170\pi\)
\(14\) 2.13138 + 1.51252i 0.569637 + 0.404239i
\(15\) 0 0
\(16\) 2.08671 − 3.41257i 0.521677 − 0.853143i
\(17\) 0.683630 + 0.394694i 0.165805 + 0.0957273i 0.580606 − 0.814185i \(-0.302816\pi\)
−0.414802 + 0.909912i \(0.636149\pi\)
\(18\) 0 0
\(19\) −2.86200 + 1.65238i −0.656589 + 0.379082i −0.790976 − 0.611847i \(-0.790427\pi\)
0.134387 + 0.990929i \(0.457093\pi\)
\(20\) −1.02000 + 0.356750i −0.228078 + 0.0797717i
\(21\) 0 0
\(22\) 2.94546 + 6.42194i 0.627973 + 1.36916i
\(23\) −3.15234 − 2.64512i −0.657307 − 0.551546i 0.251971 − 0.967735i \(-0.418921\pi\)
−0.909278 + 0.416188i \(0.863366\pi\)
\(24\) 0 0
\(25\) −4.42415 − 1.61026i −0.884830 − 0.322052i
\(26\) 1.88895 − 1.30499i 0.370453 − 0.255929i
\(27\) 0 0
\(28\) −3.45697 − 1.30783i −0.653306 − 0.247156i
\(29\) 3.24174 − 8.90660i 0.601976 − 1.65391i −0.145290 − 0.989389i \(-0.546412\pi\)
0.747266 − 0.664525i \(-0.231366\pi\)
\(30\) 0 0
\(31\) 3.70462 − 4.41499i 0.665369 − 0.792956i −0.322777 − 0.946475i \(-0.604616\pi\)
0.988146 + 0.153519i \(0.0490607\pi\)
\(32\) −1.63558 + 5.41525i −0.289132 + 0.957289i
\(33\) 0 0
\(34\) −1.07648 − 0.295728i −0.184615 − 0.0507169i
\(35\) 0.499242 + 0.864712i 0.0843873 + 0.146163i
\(36\) 0 0
\(37\) 5.53013 − 9.57846i 0.909147 − 1.57469i 0.0938963 − 0.995582i \(-0.470068\pi\)
0.815251 − 0.579108i \(-0.196599\pi\)
\(38\) 3.32552 − 3.28386i 0.539471 − 0.532712i
\(39\) 0 0
\(40\) 1.26816 − 0.852696i 0.200514 − 0.134823i
\(41\) 2.62821 + 7.22095i 0.410458 + 1.12772i 0.956948 + 0.290258i \(0.0937412\pi\)
−0.546491 + 0.837465i \(0.684037\pi\)
\(42\) 0 0
\(43\) −7.88044 + 1.38953i −1.20176 + 0.211902i −0.738458 − 0.674300i \(-0.764445\pi\)
−0.463299 + 0.886202i \(0.653334\pi\)
\(44\) −6.32553 − 7.73445i −0.953609 − 1.16601i
\(45\) 0 0
\(46\) 5.25874 + 2.49267i 0.775359 + 0.367524i
\(47\) −2.39693 + 2.01126i −0.349628 + 0.293372i −0.800640 − 0.599145i \(-0.795507\pi\)
0.451013 + 0.892517i \(0.351063\pi\)
\(48\) 0 0
\(49\) 0.622483 − 3.53027i 0.0889261 − 0.504325i
\(50\) 6.62911 + 0.622106i 0.937498 + 0.0879791i
\(51\) 0 0
\(52\) −2.11825 + 2.46074i −0.293748 + 0.341243i
\(53\) 0.933485i 0.128224i 0.997943 + 0.0641120i \(0.0204215\pi\)
−0.997943 + 0.0641120i \(0.979579\pi\)
\(54\) 0 0
\(55\) 2.69922i 0.363962i
\(56\) 5.19762 + 0.553960i 0.694561 + 0.0740260i
\(57\) 0 0
\(58\) −1.25241 + 13.3456i −0.164449 + 1.75236i
\(59\) 0.461077 − 2.61490i 0.0600271 − 0.340431i −0.939973 − 0.341250i \(-0.889150\pi\)
1.00000 0.000819421i \(0.000260830\pi\)
\(60\) 0 0
\(61\) −4.29949 + 3.60770i −0.550493 + 0.461919i −0.875108 − 0.483928i \(-0.839210\pi\)
0.324615 + 0.945846i \(0.394765\pi\)
\(62\) −3.49110 + 7.36511i −0.443370 + 0.935370i
\(63\) 0 0
\(64\) 0.302524 − 7.99428i 0.0378154 − 0.999285i
\(65\) 0.863806 − 0.152312i 0.107142 − 0.0188920i
\(66\) 0 0
\(67\) 2.15593 + 5.92336i 0.263388 + 0.723653i 0.998933 + 0.0461775i \(0.0147040\pi\)
−0.735545 + 0.677476i \(0.763074\pi\)
\(68\) 1.57865 + 0.0199049i 0.191439 + 0.00241383i
\(69\) 0 0
\(70\) −0.992169 − 1.00476i −0.118587 − 0.120092i
\(71\) −4.53203 + 7.84970i −0.537853 + 0.931588i 0.461167 + 0.887313i \(0.347431\pi\)
−0.999019 + 0.0442746i \(0.985902\pi\)
\(72\) 0 0
\(73\) 5.77108 + 9.99580i 0.675454 + 1.16992i 0.976336 + 0.216259i \(0.0693855\pi\)
−0.300882 + 0.953661i \(0.597281\pi\)
\(74\) −4.14350 + 15.0828i −0.481672 + 1.75334i
\(75\) 0 0
\(76\) −3.37666 + 5.68189i −0.387330 + 0.651757i
\(77\) −5.93457 + 7.07254i −0.676307 + 0.805991i
\(78\) 0 0
\(79\) −0.275423 + 0.756719i −0.0309875 + 0.0851375i −0.954222 − 0.299100i \(-0.903313\pi\)
0.923234 + 0.384238i \(0.125536\pi\)
\(80\) −1.43047 + 1.62000i −0.159932 + 0.181121i
\(81\) 0 0
\(82\) −6.17702 − 8.94113i −0.682137 − 0.987383i
\(83\) 15.0247 + 5.46855i 1.64918 + 0.600252i 0.988608 − 0.150513i \(-0.0480925\pi\)
0.660570 + 0.750765i \(0.270315\pi\)
\(84\) 0 0
\(85\) −0.326718 − 0.274149i −0.0354375 − 0.0297356i
\(86\) 10.2862 − 4.71783i 1.10919 − 0.508737i
\(87\) 0 0
\(88\) 11.4196 + 8.32231i 1.21733 + 0.887161i
\(89\) 5.41508 − 3.12640i 0.573997 − 0.331398i −0.184747 − 0.982786i \(-0.559147\pi\)
0.758744 + 0.651389i \(0.225813\pi\)
\(90\) 0 0
\(91\) 2.59824 + 1.50009i 0.272369 + 0.157252i
\(92\) −8.08647 − 1.53123i −0.843072 − 0.159641i
\(93\) 0 0
\(94\) 2.56089 − 3.60870i 0.264136 − 0.372208i
\(95\) 1.67785 − 0.610689i 0.172144 − 0.0626553i
\(96\) 0 0
\(97\) 1.63933 + 9.29708i 0.166448 + 0.943975i 0.947559 + 0.319582i \(0.103543\pi\)
−0.781110 + 0.624393i \(0.785346\pi\)
\(98\) 0.409997 + 5.05297i 0.0414160 + 0.510427i
\(99\) 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 324.2.l.a.179.1 96
3.2 odd 2 108.2.l.a.23.16 yes 96
4.3 odd 2 inner 324.2.l.a.179.12 96
9.2 odd 6 972.2.l.c.215.5 96
9.4 even 3 972.2.l.a.863.10 96
9.5 odd 6 972.2.l.d.863.7 96
9.7 even 3 972.2.l.b.215.12 96
12.11 even 2 108.2.l.a.23.5 ✓ 96
27.2 odd 18 972.2.l.a.107.1 96
27.7 even 9 108.2.l.a.47.5 yes 96
27.11 odd 18 972.2.l.b.755.11 96
27.16 even 9 972.2.l.c.755.6 96
27.20 odd 18 inner 324.2.l.a.143.12 96
27.25 even 9 972.2.l.d.107.16 96
36.7 odd 6 972.2.l.b.215.11 96
36.11 even 6 972.2.l.c.215.6 96
36.23 even 6 972.2.l.d.863.16 96
36.31 odd 6 972.2.l.a.863.1 96
108.7 odd 18 108.2.l.a.47.16 yes 96
108.11 even 18 972.2.l.b.755.12 96
108.43 odd 18 972.2.l.c.755.5 96
108.47 even 18 inner 324.2.l.a.143.1 96
108.79 odd 18 972.2.l.d.107.7 96
108.83 even 18 972.2.l.a.107.10 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.2.l.a.23.5 ✓ 96 12.11 even 2
108.2.l.a.23.16 yes 96 3.2 odd 2
108.2.l.a.47.5 yes 96 27.7 even 9
108.2.l.a.47.16 yes 96 108.7 odd 18
324.2.l.a.143.1 96 108.47 even 18 inner
324.2.l.a.143.12 96 27.20 odd 18 inner
324.2.l.a.179.1 96 1.1 even 1 trivial
324.2.l.a.179.12 96 4.3 odd 2 inner
972.2.l.a.107.1 96 27.2 odd 18
972.2.l.a.107.10 96 108.83 even 18
972.2.l.a.863.1 96 36.31 odd 6
972.2.l.a.863.10 96 9.4 even 3
972.2.l.b.215.11 96 36.7 odd 6
972.2.l.b.215.12 96 9.7 even 3
972.2.l.b.755.11 96 27.11 odd 18
972.2.l.b.755.12 96 108.11 even 18
972.2.l.c.215.5 96 9.2 odd 6
972.2.l.c.215.6 96 36.11 even 6
972.2.l.c.755.5 96 108.43 odd 18
972.2.l.c.755.6 96 27.16 even 9
972.2.l.d.107.7 96 108.79 odd 18
972.2.l.d.107.16 96 27.25 even 9
972.2.l.d.863.7 96 9.5 odd 6
972.2.l.d.863.16 96 36.23 even 6