↑

Properties

Label 1395.2.a.g.1.2
Level $1395$
Weight $2$
Character 1395.1
Self dual yes
Analytic conductor $11.139$
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] = [1395,2,Mod(1,1395)] 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("1395.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1395, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1395 = 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1395.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: \(11.1391310820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 465)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1395.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -0.585786 q^{7} +4.41421 q^{8} +2.41421 q^{10} +2.82843 q^{11} -2.58579 q^{13} -1.41421 q^{14} +3.00000 q^{16} +4.00000 q^{17} +2.82843 q^{19} +3.82843 q^{20} +6.82843 q^{22} +6.00000 q^{23} +1.00000 q^{25} -6.24264 q^{26} -2.24264 q^{28} -2.24264 q^{29} +1.00000 q^{31} -1.58579 q^{32} +9.65685 q^{34} -0.585786 q^{35} -1.41421 q^{37} +6.82843 q^{38} +4.41421 q^{40} +0.828427 q^{41} -11.3137 q^{43} +10.8284 q^{44} +14.4853 q^{46} +4.82843 q^{47} -6.65685 q^{49} +2.41421 q^{50} -9.89949 q^{52} +4.00000 q^{53} +2.82843 q^{55} -2.58579 q^{56} -5.41421 q^{58} +0.242641 q^{59} -10.4853 q^{61} +2.41421 q^{62} -9.82843 q^{64} -2.58579 q^{65} +3.89949 q^{67} +15.3137 q^{68} -1.41421 q^{70} -9.89949 q^{71} -5.89949 q^{73} -3.41421 q^{74} +10.8284 q^{76} -1.65685 q^{77} -14.4853 q^{79} +3.00000 q^{80} +2.00000 q^{82} +0.343146 q^{83} +4.00000 q^{85} -27.3137 q^{86} +12.4853 q^{88} -5.07107 q^{89} +1.51472 q^{91} +22.9706 q^{92} +11.6569 q^{94} +2.82843 q^{95} +15.6569 q^{97} -16.0711 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} + 6 q^{8} + 2 q^{10} - 8 q^{13} + 6 q^{16} + 8 q^{17} + 2 q^{20} + 8 q^{22} + 12 q^{23} + 2 q^{25} - 4 q^{26} + 4 q^{28} + 4 q^{29} + 2 q^{31} - 6 q^{32} + 8 q^{34}+ \cdots - 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 − 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.585786 −0.221406 −0.110703 − 0.993854i \(-0.535310\pi\)
−0.110703 + 0.993854i \(0.535310\pi\)
\(8\) 4.41421 1.56066
\(9\) 0 0
\(10\) 2.41421 0.763441
\(11\) 2.82843 0.852803 0.426401 − 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) −2.58579 −0.717168 −0.358584 − 0.933497i \(-0.616740\pi\)
−0.358584 + 0.933497i \(0.616740\pi\)
\(14\) −1.41421 −0.377964
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 − 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 3.82843 0.856062
\(21\) 0 0
\(22\) 6.82843 1.45583
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.24264 −1.22428
\(27\) 0 0
\(28\) −2.24264 −0.423819
\(29\) −2.24264 −0.416448 −0.208224 − 0.978081i \(-0.566768\pi\)
−0.208224 + 0.978081i \(0.566768\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.58579 −0.280330
\(33\) 0 0
\(34\) 9.65685 1.65614
\(35\) −0.585786 −0.0990160
\(36\) 0 0
\(37\) −1.41421 −0.232495 −0.116248 − 0.993220i \(-0.537087\pi\)
−0.116248 + 0.993220i \(0.537087\pi\)
\(38\) 6.82843 1.10772
\(39\) 0 0
\(40\) 4.41421 0.697948
\(41\) 0.828427 0.129379 0.0646893 − 0.997905i \(-0.479394\pi\)
0.0646893 + 0.997905i \(0.479394\pi\)
\(42\) 0 0
\(43\) −11.3137 −1.72532 −0.862662 − 0.505781i \(-0.831205\pi\)
−0.862662 + 0.505781i \(0.831205\pi\)
\(44\) 10.8284 1.63245
\(45\) 0 0
\(46\) 14.4853 2.13574
\(47\) 4.82843 0.704298 0.352149 − 0.935944i \(-0.385451\pi\)
0.352149 + 0.935944i \(0.385451\pi\)
\(48\) 0 0
\(49\) −6.65685 −0.950979
\(50\) 2.41421 0.341421
\(51\) 0 0
\(52\) −9.89949 −1.37281
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) −2.58579 −0.345540
\(57\) 0 0
\(58\) −5.41421 −0.710921
\(59\) 0.242641 0.0315891 0.0157946 − 0.999875i \(-0.494972\pi\)
0.0157946 + 0.999875i \(0.494972\pi\)
\(60\) 0 0
\(61\) −10.4853 −1.34250 −0.671251 − 0.741230i \(-0.734243\pi\)
−0.671251 + 0.741230i \(0.734243\pi\)
\(62\) 2.41421 0.306605
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) −2.58579 −0.320727
\(66\) 0 0
\(67\) 3.89949 0.476399 0.238200 − 0.971216i \(-0.423443\pi\)
0.238200 + 0.971216i \(0.423443\pi\)
\(68\) 15.3137 1.85706
\(69\) 0 0
\(70\) −1.41421 −0.169031
\(71\) −9.89949 −1.17485 −0.587427 − 0.809277i \(-0.699859\pi\)
−0.587427 + 0.809277i \(0.699859\pi\)
\(72\) 0 0
\(73\) −5.89949 −0.690484 −0.345242 − 0.938514i \(-0.612203\pi\)
−0.345242 + 0.938514i \(0.612203\pi\)
\(74\) −3.41421 −0.396894
\(75\) 0 0
\(76\) 10.8284 1.24211
\(77\) −1.65685 −0.188816
\(78\) 0 0
\(79\) −14.4853 −1.62972 −0.814861 − 0.579657i \(-0.803187\pi\)
−0.814861 + 0.579657i \(0.803187\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) 0.343146 0.0376651 0.0188326 − 0.999823i \(-0.494005\pi\)
0.0188326 + 0.999823i \(0.494005\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −27.3137 −2.94531
\(87\) 0 0
\(88\) 12.4853 1.33094
\(89\) −5.07107 −0.537532 −0.268766 − 0.963205i \(-0.586616\pi\)
−0.268766 + 0.963205i \(0.586616\pi\)
\(90\) 0 0
\(91\) 1.51472 0.158786
\(92\) 22.9706 2.39485
\(93\) 0 0
\(94\) 11.6569 1.20231
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) 15.6569 1.58971 0.794856 − 0.606798i \(-0.207546\pi\)
0.794856 + 0.606798i \(0.207546\pi\)
\(98\) −16.0711 −1.62342
\(99\) 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 1395.2.a.g.1.2 2
3.2 odd 2 465.2.a.c.1.1 ✓ 2
5.4 even 2 6975.2.a.u.1.1 2
12.11 even 2 7440.2.a.be.1.1 2
15.2 even 4 2325.2.c.i.1024.1 4
15.8 even 4 2325.2.c.i.1024.4 4
15.14 odd 2 2325.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.c.1.1 ✓ 2 3.2 odd 2
1395.2.a.g.1.2 2 1.1 even 1 trivial
2325.2.a.n.1.2 2 15.14 odd 2
2325.2.c.i.1024.1 4 15.2 even 4
2325.2.c.i.1024.4 4 15.8 even 4
6975.2.a.u.1.1 2 5.4 even 2
7440.2.a.be.1.1 2 12.11 even 2