↑

Properties

Label 414.2.a.e.1.1
Level $414$
Weight $2$
Character 414.1
Self dual yes
Analytic conductor $3.306$
Analytic rank $0$
Dimension $2$
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 = [2,-2,0,2,-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: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.64575 q^{5} +2.00000 q^{7} -1.00000 q^{8} +3.64575 q^{10} +3.64575 q^{11} -5.29150 q^{13} -2.00000 q^{14} +1.00000 q^{16} +7.29150 q^{17} +5.64575 q^{19} -3.64575 q^{20} -3.64575 q^{22} +1.00000 q^{23} +8.29150 q^{25} +5.29150 q^{26} +2.00000 q^{28} -1.29150 q^{29} +9.29150 q^{31} -1.00000 q^{32} -7.29150 q^{34} -7.29150 q^{35} -8.93725 q^{37} -5.64575 q^{38} +3.64575 q^{40} +6.00000 q^{41} +5.64575 q^{43} +3.64575 q^{44} -1.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} -8.29150 q^{50} -5.29150 q^{52} +3.64575 q^{53} -13.2915 q^{55} -2.00000 q^{56} +1.29150 q^{58} +5.64575 q^{61} -9.29150 q^{62} +1.00000 q^{64} +19.2915 q^{65} +0.937254 q^{67} +7.29150 q^{68} +7.29150 q^{70} -6.00000 q^{71} +3.29150 q^{73} +8.93725 q^{74} +5.64575 q^{76} +7.29150 q^{77} -12.5830 q^{79} -3.64575 q^{80} -6.00000 q^{82} -8.35425 q^{83} -26.5830 q^{85} -5.64575 q^{86} -3.64575 q^{88} -10.5830 q^{91} +1.00000 q^{92} -6.00000 q^{94} -20.5830 q^{95} +9.29150 q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 6 q^{25} + 4 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −3.64575 −1.63043 −0.815215 − 0.579159i \(-0.803381\pi\)
−0.815215 + 0.579159i \(0.803381\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.64575 1.15289
\(11\) 3.64575 1.09924 0.549618 − 0.835416i \(-0.314773\pi\)
0.549618 + 0.835416i \(0.314773\pi\)
\(12\) 0 0
\(13\) −5.29150 −1.46760 −0.733799 − 0.679366i \(-0.762255\pi\)
−0.733799 + 0.679366i \(0.762255\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.29150 1.76845 0.884225 − 0.467062i \(-0.154688\pi\)
0.884225 + 0.467062i \(0.154688\pi\)
\(18\) 0 0
\(19\) 5.64575 1.29522 0.647612 − 0.761970i \(-0.275768\pi\)
0.647612 + 0.761970i \(0.275768\pi\)
\(20\) −3.64575 −0.815215
\(21\) 0 0
\(22\) −3.64575 −0.777277
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 8.29150 1.65830
\(26\) 5.29150 1.03775
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −1.29150 −0.239826 −0.119913 − 0.992784i \(-0.538262\pi\)
−0.119913 + 0.992784i \(0.538262\pi\)
\(30\) 0 0
\(31\) 9.29150 1.66880 0.834402 − 0.551157i \(-0.185813\pi\)
0.834402 + 0.551157i \(0.185813\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −7.29150 −1.25048
\(35\) −7.29150 −1.23249
\(36\) 0 0
\(37\) −8.93725 −1.46928 −0.734638 − 0.678460i \(-0.762648\pi\)
−0.734638 + 0.678460i \(0.762648\pi\)
\(38\) −5.64575 −0.915862
\(39\) 0 0
\(40\) 3.64575 0.576444
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 5.64575 0.860969 0.430485 − 0.902598i \(-0.358343\pi\)
0.430485 + 0.902598i \(0.358343\pi\)
\(44\) 3.64575 0.549618
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −8.29150 −1.17260
\(51\) 0 0
\(52\) −5.29150 −0.733799
\(53\) 3.64575 0.500782 0.250391 − 0.968145i \(-0.419441\pi\)
0.250391 + 0.968145i \(0.419441\pi\)
\(54\) 0 0
\(55\) −13.2915 −1.79223
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 1.29150 0.169583
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 5.64575 0.722864 0.361432 − 0.932398i \(-0.382288\pi\)
0.361432 + 0.932398i \(0.382288\pi\)
\(62\) −9.29150 −1.18002
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 19.2915 2.39282
\(66\) 0 0
\(67\) 0.937254 0.114504 0.0572519 − 0.998360i \(-0.481766\pi\)
0.0572519 + 0.998360i \(0.481766\pi\)
\(68\) 7.29150 0.884225
\(69\) 0 0
\(70\) 7.29150 0.871501
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 3.29150 0.385241 0.192621 − 0.981273i \(-0.438301\pi\)
0.192621 + 0.981273i \(0.438301\pi\)
\(74\) 8.93725 1.03893
\(75\) 0 0
\(76\) 5.64575 0.647612
\(77\) 7.29150 0.830944
\(78\) 0 0
\(79\) −12.5830 −1.41570 −0.707849 − 0.706363i \(-0.750334\pi\)
−0.707849 + 0.706363i \(0.750334\pi\)
\(80\) −3.64575 −0.407607
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −8.35425 −0.916998 −0.458499 − 0.888695i \(-0.651613\pi\)
−0.458499 + 0.888695i \(0.651613\pi\)
\(84\) 0 0
\(85\) −26.5830 −2.88333
\(86\) −5.64575 −0.608797
\(87\) 0 0
\(88\) −3.64575 −0.388638
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −10.5830 −1.10940
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) −20.5830 −2.11177
\(96\) 0 0
\(97\) 9.29150 0.943409 0.471705 − 0.881757i \(-0.343639\pi\)
0.471705 + 0.881757i \(0.343639\pi\)
\(98\) 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.e.1.1 ✓ 2
3.2 odd 2 414.2.a.g.1.2 yes 2
4.3 odd 2 3312.2.a.v.1.1 2
12.11 even 2 3312.2.a.z.1.2 2
23.22 odd 2 9522.2.a.bb.1.2 2
69.68 even 2 9522.2.a.bc.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
414.2.a.e.1.1 ✓ 2 1.1 even 1 trivial
414.2.a.g.1.2 yes 2 3.2 odd 2
3312.2.a.v.1.1 2 4.3 odd 2
3312.2.a.z.1.2 2 12.11 even 2
9522.2.a.bb.1.2 2 23.22 odd 2
9522.2.a.bc.1.1 2 69.68 even 2