↑

Properties

Label 414.2.a.d.1.1
Level $414$
Weight $2$
Character 414.1
Self dual yes
Analytic conductor $3.306$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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] = [414,2,Mod(1,414)] 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("414.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(414, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,0,1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 414.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +2.00000 q^{10} +6.00000 q^{11} -2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{20} +6.00000 q^{22} +1.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} -2.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{35} +2.00000 q^{40} -10.0000 q^{41} -12.0000 q^{43} +6.00000 q^{44} +1.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} +12.0000 q^{55} -2.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} +4.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -12.0000 q^{67} -4.00000 q^{70} -10.0000 q^{73} -12.0000 q^{77} -6.00000 q^{79} +2.00000 q^{80} -10.0000 q^{82} -14.0000 q^{83} -12.0000 q^{86} +6.00000 q^{88} +4.00000 q^{91} +1.00000 q^{92} +8.00000 q^{94} -6.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 − 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) −10.0000 −1.10432
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −3.00000 −0.303046
\(99\) 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 414.2.a.d.1.1 1
3.2 odd 2 138.2.a.a.1.1 ✓ 1
4.3 odd 2 3312.2.a.n.1.1 1
12.11 even 2 1104.2.a.e.1.1 1
15.2 even 4 3450.2.d.j.2899.1 2
15.8 even 4 3450.2.d.j.2899.2 2
15.14 odd 2 3450.2.a.y.1.1 1
21.20 even 2 6762.2.a.q.1.1 1
23.22 odd 2 9522.2.a.i.1.1 1
24.5 odd 2 4416.2.a.z.1.1 1
24.11 even 2 4416.2.a.m.1.1 1
69.68 even 2 3174.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.a.1.1 ✓ 1 3.2 odd 2
414.2.a.d.1.1 1 1.1 even 1 trivial
1104.2.a.e.1.1 1 12.11 even 2
3174.2.a.b.1.1 1 69.68 even 2
3312.2.a.n.1.1 1 4.3 odd 2
3450.2.a.y.1.1 1 15.14 odd 2
3450.2.d.j.2899.1 2 15.2 even 4
3450.2.d.j.2899.2 2 15.8 even 4
4416.2.a.m.1.1 1 24.11 even 2
4416.2.a.z.1.1 1 24.5 odd 2
6762.2.a.q.1.1 1 21.20 even 2
9522.2.a.i.1.1 1 23.22 odd 2