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Properties

Label 3096.2.l.c.2321.1
Level $3096$
Weight $2$
Character 3096.2321
Analytic conductor $24.722$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [3096,2,Mod(2321,3096)] 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("3096.2321"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3096, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3096 = 2^{3} \cdot 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3096.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.7216844658\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2321.1
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 3096.2321
Dual form 3096.2.l.c.2321.2

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{5} -1.41421i q^{11} +7.07107i q^{17} +5.65685i q^{19} +1.41421i q^{23} -1.00000 q^{25} -6.00000 q^{29} -2.00000 q^{31} +8.48528i q^{37} +4.24264i q^{41} +(-5.00000 - 4.24264i) q^{43} -1.41421i q^{47} +7.00000 q^{49} +1.41421i q^{53} -2.82843i q^{55} +4.24264i q^{59} +2.82843i q^{61} +14.0000 q^{67} -12.0000 q^{71} +8.48528i q^{73} +10.0000 q^{79} -12.7279i q^{83} +14.1421i q^{85} +14.0000 q^{89} +11.3137i q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 2 q^{25} - 12 q^{29} - 4 q^{31} - 10 q^{43} + 14 q^{49} + 28 q^{67} - 24 q^{71} + 20 q^{79} + 28 q^{89} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3096\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(775\) \(1549\) \(1721\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.41421i − 0.426401i −0.977008 − 0.213201i \(-0.931611\pi\)
0.977008 − 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.07107i 1.71499i 0.514496 + 0.857493i \(0.327979\pi\)
−0.514496 + 0.857493i \(0.672021\pi\)
\(18\) 0 0
\(19\) 5.65685i 1.29777i 0.760886 + 0.648886i \(0.224765\pi\)
−0.760886 + 0.648886i \(0.775235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.48528i 1.39497i 0.716599 + 0.697486i \(0.245698\pi\)
−0.716599 + 0.697486i \(0.754302\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.24264i 0.662589i 0.943527 + 0.331295i \(0.107485\pi\)
−0.943527 + 0.331295i \(0.892515\pi\)
\(42\) 0 0
\(43\) −5.00000 − 4.24264i −0.762493 − 0.646997i
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.41421i − 0.206284i −0.994667 − 0.103142i \(-0.967110\pi\)
0.994667 − 0.103142i \(-0.0328896\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.41421i 0.194257i 0.995272 + 0.0971286i \(0.0309658\pi\)
−0.995272 + 0.0971286i \(0.969034\pi\)
\(54\) 0 0
\(55\) − 2.82843i − 0.381385i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.24264i 0.552345i 0.961108 + 0.276172i \(0.0890661\pi\)
−0.961108 + 0.276172i \(0.910934\pi\)
\(60\) 0 0
\(61\) 2.82843i 0.362143i 0.983470 + 0.181071i \(0.0579565\pi\)
−0.983470 + 0.181071i \(0.942043\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 8.48528i 0.993127i 0.868000 + 0.496564i \(0.165405\pi\)
−0.868000 + 0.496564i \(0.834595\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 12.7279i − 1.39707i −0.715575 − 0.698535i \(-0.753835\pi\)
0.715575 − 0.698535i \(-0.246165\pi\)
\(84\) 0 0
\(85\) 14.1421i 1.53393i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.3137i 1.16076i
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 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 3096.2.l.c.2321.1 yes 2
3.2 odd 2 3096.2.l.a.2321.2 yes 2
4.3 odd 2 6192.2.l.c.2321.2 2
12.11 even 2 6192.2.l.a.2321.1 2
43.42 odd 2 3096.2.l.a.2321.1 ✓ 2
129.128 even 2 inner 3096.2.l.c.2321.2 yes 2
172.171 even 2 6192.2.l.a.2321.2 2
516.515 odd 2 6192.2.l.c.2321.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3096.2.l.a.2321.1 ✓ 2 43.42 odd 2
3096.2.l.a.2321.2 yes 2 3.2 odd 2
3096.2.l.c.2321.1 yes 2 1.1 even 1 trivial
3096.2.l.c.2321.2 yes 2 129.128 even 2 inner
6192.2.l.a.2321.1 2 12.11 even 2
6192.2.l.a.2321.2 2 172.171 even 2
6192.2.l.c.2321.1 2 516.515 odd 2
6192.2.l.c.2321.2 2 4.3 odd 2