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Properties

Label 324.2.b.b.323.3
Level $324$
Weight $2$
Character 324.323
Analytic conductor $2.587$
Analytic rank $0$
Dimension $8$
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(323,324)] 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("324.323"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.170772624.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.3
Root \(0.774115 + 1.18353i\) of defining polynomial
Character \(\chi\) \(=\) 324.323
Dual form 324.2.b.b.323.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.637910 - 1.26217i) q^{2} +(-1.18614 + 1.61030i) q^{4} -0.792287i q^{5} +2.71519i q^{7} +(2.78912 + 0.469882i) q^{8} +(-1.00000 + 0.505408i) q^{10} +3.42703 q^{11} +3.37228 q^{13} +(3.42703 - 1.73205i) q^{14} +(-1.18614 - 3.82009i) q^{16} -2.52434i q^{17} +2.20979i q^{19} +(1.27582 + 0.939764i) q^{20} +(-2.18614 - 4.32550i) q^{22} +2.15121 q^{23} +4.37228 q^{25} +(-2.15121 - 4.25639i) q^{26} +(-4.37228 - 3.22060i) q^{28} -0.792287i q^{29} +1.70438i q^{31} +(-4.06494 + 3.93398i) q^{32} +(-3.18614 + 1.61030i) q^{34} +2.15121 q^{35} +4.74456 q^{37} +(2.78912 - 1.40965i) q^{38} +(0.372281 - 2.20979i) q^{40} -0.147477i q^{41} -6.94661i q^{43} +(-4.06494 + 5.51856i) q^{44} +(-1.37228 - 2.71519i) q^{46} -11.5569 q^{47} -0.372281 q^{49} +(-2.78912 - 5.51856i) q^{50} +(-4.00000 + 5.43039i) q^{52} +8.51278i q^{53} -2.71519i q^{55} +(-1.27582 + 7.57301i) q^{56} +(-1.00000 + 0.505408i) q^{58} -5.17782 q^{59} +3.37228 q^{61} +(2.15121 - 1.08724i) q^{62} +(7.55842 + 2.62112i) q^{64} -2.67181i q^{65} +6.94661i q^{67} +(4.06494 + 2.99422i) q^{68} +(-1.37228 - 2.71519i) q^{70} -1.75079 q^{71} -2.37228 q^{73} +(-3.02661 - 5.98844i) q^{74} +(-3.55842 - 2.62112i) q^{76} +9.30506i q^{77} +10.1672i q^{79} +(-3.02661 + 0.939764i) q^{80} +(-0.186141 + 0.0940770i) q^{82} +7.25450 q^{83} -2.00000 q^{85} +(-8.76780 + 4.43132i) q^{86} +(9.55842 + 1.61030i) q^{88} -5.34363i q^{89} +9.15640i q^{91} +(-2.55164 + 3.46410i) q^{92} +(7.37228 + 14.5868i) q^{94} +1.75079 q^{95} -12.4891 q^{97} +(0.237482 + 0.469882i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 8 q^{10} + 4 q^{13} + 2 q^{16} - 6 q^{22} + 12 q^{25} - 12 q^{28} - 14 q^{34} - 8 q^{37} - 20 q^{40} + 12 q^{46} + 20 q^{49} - 32 q^{52} - 8 q^{58} + 4 q^{61} + 26 q^{64} + 12 q^{70} + 4 q^{73}+ \cdots - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.637910 − 1.26217i −0.451071 − 0.892488i
\(3\) 0 0
\(4\) −1.18614 + 1.61030i −0.593070 + 0.805151i
\(5\) − 0.792287i − 0.354322i −0.984182 − 0.177161i \(-0.943309\pi\)
0.984182 − 0.177161i \(-0.0566913\pi\)
\(6\) 0 0
\(7\) 2.71519i 1.02625i 0.858315 + 0.513124i \(0.171512\pi\)
−0.858315 + 0.513124i \(0.828488\pi\)
\(8\) 2.78912 + 0.469882i 0.986104 + 0.166128i
\(9\) 0 0
\(10\) −1.00000 + 0.505408i −0.316228 + 0.159824i
\(11\) 3.42703 1.03329 0.516645 − 0.856200i \(-0.327181\pi\)
0.516645 + 0.856200i \(0.327181\pi\)
\(12\) 0 0
\(13\) 3.37228 0.935303 0.467651 − 0.883913i \(-0.345100\pi\)
0.467651 + 0.883913i \(0.345100\pi\)
\(14\) 3.42703 − 1.73205i 0.915913 − 0.462910i
\(15\) 0 0
\(16\) −1.18614 − 3.82009i −0.296535 − 0.955022i
\(17\) − 2.52434i − 0.612242i −0.951993 − 0.306121i \(-0.900969\pi\)
0.951993 − 0.306121i \(-0.0990312\pi\)
\(18\) 0 0
\(19\) 2.20979i 0.506960i 0.967341 + 0.253480i \(0.0815752\pi\)
−0.967341 + 0.253480i \(0.918425\pi\)
\(20\) 1.27582 + 0.939764i 0.285282 + 0.210138i
\(21\) 0 0
\(22\) −2.18614 − 4.32550i −0.466087 − 0.922199i
\(23\) 2.15121 0.448559 0.224279 − 0.974525i \(-0.427997\pi\)
0.224279 + 0.974525i \(0.427997\pi\)
\(24\) 0 0
\(25\) 4.37228 0.874456
\(26\) −2.15121 − 4.25639i −0.421888 − 0.834746i
\(27\) 0 0
\(28\) −4.37228 − 3.22060i −0.826284 − 0.608637i
\(29\) − 0.792287i − 0.147124i −0.997291 − 0.0735620i \(-0.976563\pi\)
0.997291 − 0.0735620i \(-0.0234367\pi\)
\(30\) 0 0
\(31\) 1.70438i 0.306115i 0.988217 + 0.153058i \(0.0489120\pi\)
−0.988217 + 0.153058i \(0.951088\pi\)
\(32\) −4.06494 + 3.93398i −0.718587 + 0.695437i
\(33\) 0 0
\(34\) −3.18614 + 1.61030i −0.546419 + 0.276164i
\(35\) 2.15121 0.363621
\(36\) 0 0
\(37\) 4.74456 0.780001 0.390001 − 0.920815i \(-0.372475\pi\)
0.390001 + 0.920815i \(0.372475\pi\)
\(38\) 2.78912 − 1.40965i 0.452456 − 0.228675i
\(39\) 0 0
\(40\) 0.372281 − 2.20979i 0.0588628 − 0.349398i
\(41\) − 0.147477i − 0.0230320i −0.999934 − 0.0115160i \(-0.996334\pi\)
0.999934 − 0.0115160i \(-0.00366574\pi\)
\(42\) 0 0
\(43\) − 6.94661i − 1.05935i −0.848201 − 0.529674i \(-0.822314\pi\)
0.848201 − 0.529674i \(-0.177686\pi\)
\(44\) −4.06494 + 5.51856i −0.612813 + 0.831954i
\(45\) 0 0
\(46\) −1.37228 − 2.71519i −0.202332 − 0.400334i
\(47\) −11.5569 −1.68575 −0.842875 − 0.538109i \(-0.819139\pi\)
−0.842875 + 0.538109i \(0.819139\pi\)
\(48\) 0 0
\(49\) −0.372281 −0.0531830
\(50\) −2.78912 − 5.51856i −0.394442 − 0.780442i
\(51\) 0 0
\(52\) −4.00000 + 5.43039i −0.554700 + 0.753059i
\(53\) 8.51278i 1.16932i 0.811278 + 0.584660i \(0.198772\pi\)
−0.811278 + 0.584660i \(0.801228\pi\)
\(54\) 0 0
\(55\) − 2.71519i − 0.366117i
\(56\) −1.27582 + 7.57301i −0.170489 + 1.01199i
\(57\) 0 0
\(58\) −1.00000 + 0.505408i −0.131306 + 0.0663633i
\(59\) −5.17782 −0.674095 −0.337047 − 0.941488i \(-0.609428\pi\)
−0.337047 + 0.941488i \(0.609428\pi\)
\(60\) 0 0
\(61\) 3.37228 0.431776 0.215888 − 0.976418i \(-0.430735\pi\)
0.215888 + 0.976418i \(0.430735\pi\)
\(62\) 2.15121 − 1.08724i 0.273204 − 0.138080i
\(63\) 0 0
\(64\) 7.55842 + 2.62112i 0.944803 + 0.327640i
\(65\) − 2.67181i − 0.331398i
\(66\) 0 0
\(67\) 6.94661i 0.848664i 0.905507 + 0.424332i \(0.139491\pi\)
−0.905507 + 0.424332i \(0.860509\pi\)
\(68\) 4.06494 + 2.99422i 0.492947 + 0.363102i
\(69\) 0 0
\(70\) −1.37228 − 2.71519i −0.164019 − 0.324528i
\(71\) −1.75079 −0.207780 −0.103890 − 0.994589i \(-0.533129\pi\)
−0.103890 + 0.994589i \(0.533129\pi\)
\(72\) 0 0
\(73\) −2.37228 −0.277655 −0.138827 − 0.990317i \(-0.544333\pi\)
−0.138827 + 0.990317i \(0.544333\pi\)
\(74\) −3.02661 − 5.98844i −0.351836 − 0.696142i
\(75\) 0 0
\(76\) −3.55842 − 2.62112i −0.408179 − 0.300663i
\(77\) 9.30506i 1.06041i
\(78\) 0 0
\(79\) 10.1672i 1.14390i 0.820288 + 0.571951i \(0.193813\pi\)
−0.820288 + 0.571951i \(0.806187\pi\)
\(80\) −3.02661 + 0.939764i −0.338385 + 0.105069i
\(81\) 0 0
\(82\) −0.186141 + 0.0940770i −0.0205558 + 0.0103891i
\(83\) 7.25450 0.796284 0.398142 − 0.917324i \(-0.369655\pi\)
0.398142 + 0.917324i \(0.369655\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −8.76780 + 4.43132i −0.945456 + 0.477841i
\(87\) 0 0
\(88\) 9.55842 + 1.61030i 1.01893 + 0.171659i
\(89\) − 5.34363i − 0.566424i −0.959057 − 0.283212i \(-0.908600\pi\)
0.959057 − 0.283212i \(-0.0913999\pi\)
\(90\) 0 0
\(91\) 9.15640i 0.959852i
\(92\) −2.55164 + 3.46410i −0.266027 + 0.361158i
\(93\) 0 0
\(94\) 7.37228 + 14.5868i 0.760393 + 1.50451i
\(95\) 1.75079 0.179627
\(96\) 0 0
\(97\) −12.4891 −1.26808 −0.634039 − 0.773301i \(-0.718604\pi\)
−0.634039 + 0.773301i \(0.718604\pi\)
\(98\) 0.237482 + 0.469882i 0.0239893 + 0.0474652i
\(99\) 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.b.b.323.3 8
3.2 odd 2 inner 324.2.b.b.323.6 8
4.3 odd 2 inner 324.2.b.b.323.5 8
8.3 odd 2 5184.2.c.j.5183.5 8
8.5 even 2 5184.2.c.j.5183.6 8
9.2 odd 6 108.2.h.a.71.3 8
9.4 even 3 108.2.h.a.35.4 8
9.5 odd 6 36.2.h.a.11.1 ✓ 8
9.7 even 3 36.2.h.a.23.2 yes 8
12.11 even 2 inner 324.2.b.b.323.4 8
24.5 odd 2 5184.2.c.j.5183.4 8
24.11 even 2 5184.2.c.j.5183.3 8
36.7 odd 6 36.2.h.a.23.1 yes 8
36.11 even 6 108.2.h.a.71.4 8
36.23 even 6 36.2.h.a.11.2 yes 8
36.31 odd 6 108.2.h.a.35.3 8
45.7 odd 12 900.2.o.a.599.2 16
45.14 odd 6 900.2.r.c.551.4 8
45.23 even 12 900.2.o.a.299.4 16
45.32 even 12 900.2.o.a.299.5 16
45.34 even 6 900.2.r.c.851.3 8
45.43 odd 12 900.2.o.a.599.7 16
72.5 odd 6 576.2.s.f.191.2 8
72.11 even 6 1728.2.s.f.1151.3 8
72.13 even 6 1728.2.s.f.575.3 8
72.29 odd 6 1728.2.s.f.1151.4 8
72.43 odd 6 576.2.s.f.383.2 8
72.59 even 6 576.2.s.f.191.3 8
72.61 even 6 576.2.s.f.383.3 8
72.67 odd 6 1728.2.s.f.575.4 8
180.7 even 12 900.2.o.a.599.4 16
180.23 odd 12 900.2.o.a.299.2 16
180.43 even 12 900.2.o.a.599.5 16
180.59 even 6 900.2.r.c.551.3 8
180.79 odd 6 900.2.r.c.851.4 8
180.167 odd 12 900.2.o.a.299.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.2.h.a.11.1 ✓ 8 9.5 odd 6
36.2.h.a.11.2 yes 8 36.23 even 6
36.2.h.a.23.1 yes 8 36.7 odd 6
36.2.h.a.23.2 yes 8 9.7 even 3
108.2.h.a.35.3 8 36.31 odd 6
108.2.h.a.35.4 8 9.4 even 3
108.2.h.a.71.3 8 9.2 odd 6
108.2.h.a.71.4 8 36.11 even 6
324.2.b.b.323.3 8 1.1 even 1 trivial
324.2.b.b.323.4 8 12.11 even 2 inner
324.2.b.b.323.5 8 4.3 odd 2 inner
324.2.b.b.323.6 8 3.2 odd 2 inner
576.2.s.f.191.2 8 72.5 odd 6
576.2.s.f.191.3 8 72.59 even 6
576.2.s.f.383.2 8 72.43 odd 6
576.2.s.f.383.3 8 72.61 even 6
900.2.o.a.299.2 16 180.23 odd 12
900.2.o.a.299.4 16 45.23 even 12
900.2.o.a.299.5 16 45.32 even 12
900.2.o.a.299.7 16 180.167 odd 12
900.2.o.a.599.2 16 45.7 odd 12
900.2.o.a.599.4 16 180.7 even 12
900.2.o.a.599.5 16 180.43 even 12
900.2.o.a.599.7 16 45.43 odd 12
900.2.r.c.551.3 8 180.59 even 6
900.2.r.c.551.4 8 45.14 odd 6
900.2.r.c.851.3 8 45.34 even 6
900.2.r.c.851.4 8 180.79 odd 6
1728.2.s.f.575.3 8 72.13 even 6
1728.2.s.f.575.4 8 72.67 odd 6
1728.2.s.f.1151.3 8 72.11 even 6
1728.2.s.f.1151.4 8 72.29 odd 6
5184.2.c.j.5183.3 8 24.11 even 2
5184.2.c.j.5183.4 8 24.5 odd 2
5184.2.c.j.5183.5 8 8.3 odd 2
5184.2.c.j.5183.6 8 8.5 even 2