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Properties

Label 12.8.a.a.1.1
Level $12$
Weight $8$
Character 12.1
Self dual yes
Analytic conductor $3.749$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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] = [12,8,Mod(1,12)] 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("12.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 12.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-27] 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: yes
Analytic conductor: \(3.74862030581\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 12.1

$q$-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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -378.000 q^{5} -832.000 q^{7} +729.000 q^{9} -2484.00 q^{11} +14870.0 q^{13} +10206.0 q^{15} -22302.0 q^{17} -16300.0 q^{19} +22464.0 q^{21} -115128. q^{23} +64759.0 q^{25} -19683.0 q^{27} +157086. q^{29} -16456.0 q^{31} +67068.0 q^{33} +314496. q^{35} -149266. q^{37} -401490. q^{39} -241110. q^{41} -443188. q^{43} -275562. q^{45} +922752. q^{47} -131319. q^{49} +602154. q^{51} -697626. q^{53} +938952. q^{55} +440100. q^{57} +870156. q^{59} +2.06706e6 q^{61} -606528. q^{63} -5.62086e6 q^{65} -1.68075e6 q^{67} +3.10846e6 q^{69} -1.07028e6 q^{71} -2.40333e6 q^{73} -1.74849e6 q^{75} +2.06669e6 q^{77} +2.30151e6 q^{79} +531441. q^{81} +4.70869e6 q^{83} +8.43016e6 q^{85} -4.24132e6 q^{87} +4.14369e6 q^{89} -1.23718e7 q^{91} +444312. q^{93} +6.16140e6 q^{95} -1.62297e6 q^{97} -1.81084e6 q^{99} +O(q^{100})\)

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
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) −378.000 −1.35237 −0.676187 − 0.736730i \(-0.736369\pi\)
−0.676187 + 0.736730i \(0.736369\pi\)
\(6\) 0 0
\(7\) −832.000 −0.916812 −0.458406 − 0.888743i \(-0.651579\pi\)
−0.458406 + 0.888743i \(0.651579\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −2484.00 −0.562700 −0.281350 − 0.959605i \(-0.590782\pi\)
−0.281350 + 0.959605i \(0.590782\pi\)
\(12\) 0 0
\(13\) 14870.0 1.87719 0.938597 − 0.345015i \(-0.112126\pi\)
0.938597 + 0.345015i \(0.112126\pi\)
\(14\) 0 0
\(15\) 10206.0 0.780793
\(16\) 0 0
\(17\) −22302.0 −1.10096 −0.550481 − 0.834847i \(-0.685556\pi\)
−0.550481 + 0.834847i \(0.685556\pi\)
\(18\) 0 0
\(19\) −16300.0 −0.545193 −0.272596 − 0.962128i \(-0.587882\pi\)
−0.272596 + 0.962128i \(0.587882\pi\)
\(20\) 0 0
\(21\) 22464.0 0.529322
\(22\) 0 0
\(23\) −115128. −1.97303 −0.986515 − 0.163673i \(-0.947666\pi\)
−0.986515 + 0.163673i \(0.947666\pi\)
\(24\) 0 0
\(25\) 64759.0 0.828915
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) 157086. 1.19604 0.598018 − 0.801482i \(-0.295955\pi\)
0.598018 + 0.801482i \(0.295955\pi\)
\(30\) 0 0
\(31\) −16456.0 −0.0992107 −0.0496053 − 0.998769i \(-0.515796\pi\)
−0.0496053 + 0.998769i \(0.515796\pi\)
\(32\) 0 0
\(33\) 67068.0 0.324875
\(34\) 0 0
\(35\) 314496. 1.23987
\(36\) 0 0
\(37\) −149266. −0.484457 −0.242228 − 0.970219i \(-0.577878\pi\)
−0.242228 + 0.970219i \(0.577878\pi\)
\(38\) 0 0
\(39\) −401490. −1.08380
\(40\) 0 0
\(41\) −241110. −0.546351 −0.273175 − 0.961964i \(-0.588074\pi\)
−0.273175 + 0.961964i \(0.588074\pi\)
\(42\) 0 0
\(43\) −443188. −0.850058 −0.425029 − 0.905180i \(-0.639736\pi\)
−0.425029 + 0.905180i \(0.639736\pi\)
\(44\) 0 0
\(45\) −275562. −0.450791
\(46\) 0 0
\(47\) 922752. 1.29641 0.648205 − 0.761466i \(-0.275520\pi\)
0.648205 + 0.761466i \(0.275520\pi\)
\(48\) 0 0
\(49\) −131319. −0.159456
\(50\) 0 0
\(51\) 602154. 0.635641
\(52\) 0 0
\(53\) −697626. −0.643661 −0.321830 − 0.946797i \(-0.604298\pi\)
−0.321830 + 0.946797i \(0.604298\pi\)
\(54\) 0 0
\(55\) 938952. 0.760981
\(56\) 0 0
\(57\) 440100. 0.314767
\(58\) 0 0
\(59\) 870156. 0.551588 0.275794 − 0.961217i \(-0.411059\pi\)
0.275794 + 0.961217i \(0.411059\pi\)
\(60\) 0 0
\(61\) 2.06706e6 1.16600 0.583001 − 0.812472i \(-0.301878\pi\)
0.583001 + 0.812472i \(0.301878\pi\)
\(62\) 0 0
\(63\) −606528. −0.305604
\(64\) 0 0
\(65\) −5.62086e6 −2.53867
\(66\) 0 0
\(67\) −1.68075e6 −0.682717 −0.341359 − 0.939933i \(-0.610887\pi\)
−0.341359 + 0.939933i \(0.610887\pi\)
\(68\) 0 0
\(69\) 3.10846e6 1.13913
\(70\) 0 0
\(71\) −1.07028e6 −0.354890 −0.177445 − 0.984131i \(-0.556783\pi\)
−0.177445 + 0.984131i \(0.556783\pi\)
\(72\) 0 0
\(73\) −2.40333e6 −0.723076 −0.361538 − 0.932357i \(-0.617748\pi\)
−0.361538 + 0.932357i \(0.617748\pi\)
\(74\) 0 0
\(75\) −1.74849e6 −0.478574
\(76\) 0 0
\(77\) 2.06669e6 0.515890
\(78\) 0 0
\(79\) 2.30151e6 0.525192 0.262596 − 0.964906i \(-0.415421\pi\)
0.262596 + 0.964906i \(0.415421\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 4.70869e6 0.903914 0.451957 − 0.892040i \(-0.350726\pi\)
0.451957 + 0.892040i \(0.350726\pi\)
\(84\) 0 0
\(85\) 8.43016e6 1.48891
\(86\) 0 0
\(87\) −4.24132e6 −0.690532
\(88\) 0 0
\(89\) 4.14369e6 0.623049 0.311525 − 0.950238i \(-0.399160\pi\)
0.311525 + 0.950238i \(0.399160\pi\)
\(90\) 0 0
\(91\) −1.23718e7 −1.72103
\(92\) 0 0
\(93\) 444312. 0.0572793
\(94\) 0 0
\(95\) 6.16140e6 0.737304
\(96\) 0 0
\(97\) −1.62297e6 −0.180555 −0.0902777 − 0.995917i \(-0.528775\pi\)
−0.0902777 + 0.995917i \(0.528775\pi\)
\(98\) 0 0
\(99\) −1.81084e6 −0.187567
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 12.8.a.a.1.1 ✓ 1
3.2 odd 2 36.8.a.c.1.1 1
4.3 odd 2 48.8.a.e.1.1 1
5.2 odd 4 300.8.d.c.49.2 2
5.3 odd 4 300.8.d.c.49.1 2
5.4 even 2 300.8.a.g.1.1 1
7.2 even 3 588.8.i.h.361.1 2
7.3 odd 6 588.8.i.a.373.1 2
7.4 even 3 588.8.i.h.373.1 2
7.5 odd 6 588.8.i.a.361.1 2
7.6 odd 2 588.8.a.d.1.1 1
8.3 odd 2 192.8.a.g.1.1 1
8.5 even 2 192.8.a.o.1.1 1
9.2 odd 6 324.8.e.a.109.1 2
9.4 even 3 324.8.e.f.217.1 2
9.5 odd 6 324.8.e.a.217.1 2
9.7 even 3 324.8.e.f.109.1 2
12.11 even 2 144.8.a.j.1.1 1
24.5 odd 2 576.8.a.d.1.1 1
24.11 even 2 576.8.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.8.a.a.1.1 ✓ 1 1.1 even 1 trivial
36.8.a.c.1.1 1 3.2 odd 2
48.8.a.e.1.1 1 4.3 odd 2
144.8.a.j.1.1 1 12.11 even 2
192.8.a.g.1.1 1 8.3 odd 2
192.8.a.o.1.1 1 8.5 even 2
300.8.a.g.1.1 1 5.4 even 2
300.8.d.c.49.1 2 5.3 odd 4
300.8.d.c.49.2 2 5.2 odd 4
324.8.e.a.109.1 2 9.2 odd 6
324.8.e.a.217.1 2 9.5 odd 6
324.8.e.f.109.1 2 9.7 even 3
324.8.e.f.217.1 2 9.4 even 3
576.8.a.d.1.1 1 24.5 odd 2
576.8.a.e.1.1 1 24.11 even 2
588.8.a.d.1.1 1 7.6 odd 2
588.8.i.a.361.1 2 7.5 odd 6
588.8.i.a.373.1 2 7.3 odd 6
588.8.i.h.361.1 2 7.2 even 3
588.8.i.h.373.1 2 7.4 even 3