Properties

Label 2916.2.a.e.1.4
Level $2916$
Weight $2$
Character 2916.1
Self dual yes
Analytic conductor $23.284$
Analytic rank $0$
Dimension $12$
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] = [2916,2,Mod(1,2916)] 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("2916.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2916, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2916 = 2^{2} \cdot 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2916.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0] 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: \(23.2843772294\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 198x^{8} - 734x^{6} + 1329x^{4} - 1134x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.916904\) of defining polynomial
Character \(\chi\) \(=\) 2916.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.32168 q^{5} +3.12021 q^{7} +3.13990 q^{11} -3.51678 q^{13} +5.62105 q^{17} +5.22552 q^{19} -2.03388 q^{23} +0.390204 q^{25} -6.17841 q^{29} +4.83648 q^{31} -7.24414 q^{35} +1.75386 q^{37} -11.7953 q^{41} +8.82295 q^{43} -5.48809 q^{47} +2.73573 q^{49} -5.46571 q^{53} -7.28984 q^{55} +9.86189 q^{59} +10.8499 q^{61} +8.16485 q^{65} +6.57813 q^{67} +3.25445 q^{71} -5.61431 q^{73} +9.79715 q^{77} +15.7453 q^{79} +13.9214 q^{83} -13.0503 q^{85} -7.16926 q^{89} -10.9731 q^{91} -12.1320 q^{95} +13.5392 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{13} + 12 q^{19} + 30 q^{25} + 6 q^{31} + 18 q^{37} + 24 q^{43} + 48 q^{49} + 18 q^{55} + 60 q^{61} + 42 q^{67} + 60 q^{73} + 12 q^{79} + 72 q^{85} + 48 q^{91} + 66 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.32168 −1.03829 −0.519144 − 0.854687i \(-0.673749\pi\)
−0.519144 + 0.854687i \(0.673749\pi\)
\(6\) 0 0
\(7\) 3.12021 1.17933 0.589665 − 0.807648i \(-0.299260\pi\)
0.589665 + 0.807648i \(0.299260\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.13990 0.946715 0.473357 − 0.880871i \(-0.343042\pi\)
0.473357 + 0.880871i \(0.343042\pi\)
\(12\) 0 0
\(13\) −3.51678 −0.975381 −0.487690 − 0.873017i \(-0.662160\pi\)
−0.487690 + 0.873017i \(0.662160\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.62105 1.36330 0.681652 − 0.731676i \(-0.261262\pi\)
0.681652 + 0.731676i \(0.261262\pi\)
\(18\) 0 0
\(19\) 5.22552 1.19882 0.599409 − 0.800443i \(-0.295402\pi\)
0.599409 + 0.800443i \(0.295402\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.03388 −0.424094 −0.212047 − 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(24\) 0 0
\(25\) 0.390204 0.0780408
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.17841 −1.14730 −0.573651 − 0.819100i \(-0.694473\pi\)
−0.573651 + 0.819100i \(0.694473\pi\)
\(30\) 0 0
\(31\) 4.83648 0.868657 0.434328 − 0.900755i \(-0.356986\pi\)
0.434328 + 0.900755i \(0.356986\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.24414 −1.22448
\(36\) 0 0
\(37\) 1.75386 0.288332 0.144166 − 0.989553i \(-0.453950\pi\)
0.144166 + 0.989553i \(0.453950\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.7953 −1.84212 −0.921060 − 0.389421i \(-0.872675\pi\)
−0.921060 + 0.389421i \(0.872675\pi\)
\(42\) 0 0
\(43\) 8.82295 1.34549 0.672743 − 0.739876i \(-0.265116\pi\)
0.672743 + 0.739876i \(0.265116\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.48809 −0.800520 −0.400260 − 0.916402i \(-0.631080\pi\)
−0.400260 + 0.916402i \(0.631080\pi\)
\(48\) 0 0
\(49\) 2.73573 0.390818
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.46571 −0.750773 −0.375387 − 0.926868i \(-0.622490\pi\)
−0.375387 + 0.926868i \(0.622490\pi\)
\(54\) 0 0
\(55\) −7.28984 −0.982962
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.86189 1.28391 0.641954 − 0.766743i \(-0.278124\pi\)
0.641954 + 0.766743i \(0.278124\pi\)
\(60\) 0 0
\(61\) 10.8499 1.38919 0.694596 − 0.719400i \(-0.255583\pi\)
0.694596 + 0.719400i \(0.255583\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.16485 1.01273
\(66\) 0 0
\(67\) 6.57813 0.803647 0.401824 − 0.915717i \(-0.368377\pi\)
0.401824 + 0.915717i \(0.368377\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.25445 0.386232 0.193116 − 0.981176i \(-0.438141\pi\)
0.193116 + 0.981176i \(0.438141\pi\)
\(72\) 0 0
\(73\) −5.61431 −0.657105 −0.328552 − 0.944486i \(-0.606561\pi\)
−0.328552 + 0.944486i \(0.606561\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.79715 1.11649
\(78\) 0 0
\(79\) 15.7453 1.77148 0.885741 − 0.464180i \(-0.153651\pi\)
0.885741 + 0.464180i \(0.153651\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.9214 1.52807 0.764035 − 0.645175i \(-0.223215\pi\)
0.764035 + 0.645175i \(0.223215\pi\)
\(84\) 0 0
\(85\) −13.0503 −1.41550
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.16926 −0.759940 −0.379970 − 0.924999i \(-0.624066\pi\)
−0.379970 + 0.924999i \(0.624066\pi\)
\(90\) 0 0
\(91\) −10.9731 −1.15030
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −12.1320 −1.24472
\(96\) 0 0
\(97\) 13.5392 1.37470 0.687349 − 0.726327i \(-0.258774\pi\)
0.687349 + 0.726327i \(0.258774\pi\)
\(98\) 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 2916.2.a.e.1.4 ✓ 12
3.2 odd 2 inner 2916.2.a.e.1.9 yes 12
9.2 odd 6 2916.2.e.e.973.4 24
9.4 even 3 2916.2.e.e.1945.9 24
9.5 odd 6 2916.2.e.e.1945.4 24
9.7 even 3 2916.2.e.e.973.9 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2916.2.a.e.1.4 ✓ 12 1.1 even 1 trivial
2916.2.a.e.1.9 yes 12 3.2 odd 2 inner
2916.2.e.e.973.4 24 9.2 odd 6
2916.2.e.e.973.9 24 9.7 even 3
2916.2.e.e.1945.4 24 9.5 odd 6
2916.2.e.e.1945.9 24 9.4 even 3