Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1232,2,Mod(177,1232)] 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("1232.177"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1232, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,0,4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 177.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1232.177
Dual form 1232.2.q.f.529.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.20711 - 2.09077i) q^{3} +(0.292893 - 0.507306i) q^{5} +(-1.62132 - 2.09077i) q^{7} +(-1.41421 + 2.44949i) q^{9} +(0.500000 + 0.866025i) q^{11} -3.82843 q^{13} -1.41421 q^{15} +(-1.82843 - 3.16693i) q^{17} +(-0.292893 + 0.507306i) q^{19} +(-2.41421 + 5.91359i) q^{21} +(-3.12132 + 5.40629i) q^{23} +(2.32843 + 4.03295i) q^{25} -0.414214 q^{27} +2.65685 q^{29} +(-2.00000 - 3.46410i) q^{31} +(1.20711 - 2.09077i) q^{33} +(-1.53553 + 0.210133i) q^{35} +(4.70711 - 8.15295i) q^{37} +(4.62132 + 8.00436i) q^{39} -5.41421 q^{41} +5.65685 q^{43} +(0.828427 + 1.43488i) q^{45} +(-5.24264 + 9.08052i) q^{47} +(-1.74264 + 6.77962i) q^{49} +(-4.41421 + 7.64564i) q^{51} +(-3.94975 - 6.84116i) q^{53} +0.585786 q^{55} +1.41421 q^{57} +(-2.79289 - 4.83743i) q^{59} +(-5.91421 + 10.2437i) q^{61} +(7.41421 - 1.01461i) q^{63} +(-1.12132 + 1.94218i) q^{65} +(1.37868 + 2.38794i) q^{67} +15.0711 q^{69} +11.0711 q^{71} +(4.70711 + 8.15295i) q^{73} +(5.62132 - 9.73641i) q^{75} +(1.00000 - 2.44949i) q^{77} +(-6.62132 + 11.4685i) q^{79} +(4.74264 + 8.21449i) q^{81} +12.1421 q^{83} -2.14214 q^{85} +(-3.20711 - 5.55487i) q^{87} +(-6.24264 + 10.8126i) q^{89} +(6.20711 + 8.00436i) q^{91} +(-4.82843 + 8.36308i) q^{93} +(0.171573 + 0.297173i) q^{95} -3.82843 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{17} - 4 q^{19} - 4 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} - 12 q^{29} - 8 q^{31} + 2 q^{33} + 8 q^{35} + 16 q^{37} + 10 q^{39} - 16 q^{41}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.20711 − 2.09077i −0.696923 − 1.20711i −0.969528 − 0.244981i \(-0.921218\pi\)
0.272605 − 0.962126i \(-0.412115\pi\)
\(4\) 0 0
\(5\) 0.292893 − 0.507306i 0.130986 − 0.226874i −0.793071 − 0.609129i \(-0.791519\pi\)
0.924057 + 0.382255i \(0.124852\pi\)
\(6\) 0 0
\(7\) −1.62132 − 2.09077i −0.612801 − 0.790237i
\(8\) 0 0
\(9\) −1.41421 + 2.44949i −0.471405 + 0.816497i
\(10\) 0 0
\(11\) 0.500000 + 0.866025i 0.150756 + 0.261116i
\(12\) 0 0
\(13\) −3.82843 −1.06181 −0.530907 − 0.847430i \(-0.678149\pi\)
−0.530907 + 0.847430i \(0.678149\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) −1.82843 − 3.16693i −0.443459 − 0.768093i 0.554485 − 0.832194i \(-0.312915\pi\)
−0.997943 + 0.0641009i \(0.979582\pi\)
\(18\) 0 0
\(19\) −0.292893 + 0.507306i −0.0671943 + 0.116384i −0.897665 − 0.440678i \(-0.854738\pi\)
0.830471 + 0.557062i \(0.188071\pi\)
\(20\) 0 0
\(21\) −2.41421 + 5.91359i −0.526825 + 1.29045i
\(22\) 0 0
\(23\) −3.12132 + 5.40629i −0.650840 + 1.12729i 0.332079 + 0.943252i \(0.392250\pi\)
−0.982919 + 0.184037i \(0.941083\pi\)
\(24\) 0 0
\(25\) 2.32843 + 4.03295i 0.465685 + 0.806591i
\(26\) 0 0
\(27\) −0.414214 −0.0797154
\(28\) 0 0
\(29\) 2.65685 0.493365 0.246683 − 0.969096i \(-0.420659\pi\)
0.246683 + 0.969096i \(0.420659\pi\)
\(30\) 0 0
\(31\) −2.00000 − 3.46410i −0.359211 − 0.622171i 0.628619 − 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 1.20711 − 2.09077i 0.210130 − 0.363956i
\(34\) 0 0
\(35\) −1.53553 + 0.210133i −0.259553 + 0.0355190i
\(36\) 0 0
\(37\) 4.70711 − 8.15295i 0.773844 − 1.34034i −0.161599 − 0.986857i \(-0.551665\pi\)
0.935442 − 0.353480i \(-0.115002\pi\)
\(38\) 0 0
\(39\) 4.62132 + 8.00436i 0.740003 + 1.28172i
\(40\) 0 0
\(41\) −5.41421 −0.845558 −0.422779 − 0.906233i \(-0.638945\pi\)
−0.422779 + 0.906233i \(0.638945\pi\)
\(42\) 0 0
\(43\) 5.65685 0.862662 0.431331 − 0.902194i \(-0.358044\pi\)
0.431331 + 0.902194i \(0.358044\pi\)
\(44\) 0 0
\(45\) 0.828427 + 1.43488i 0.123495 + 0.213899i
\(46\) 0 0
\(47\) −5.24264 + 9.08052i −0.764718 + 1.32453i 0.175678 + 0.984448i \(0.443788\pi\)
−0.940396 + 0.340082i \(0.889545\pi\)
\(48\) 0 0
\(49\) −1.74264 + 6.77962i −0.248949 + 0.968517i
\(50\) 0 0
\(51\) −4.41421 + 7.64564i −0.618114 + 1.07060i
\(52\) 0 0
\(53\) −3.94975 − 6.84116i −0.542540 − 0.939706i −0.998757 − 0.0498379i \(-0.984130\pi\)
0.456218 − 0.889868i \(-0.349204\pi\)
\(54\) 0 0
\(55\) 0.585786 0.0789874
\(56\) 0 0
\(57\) 1.41421 0.187317
\(58\) 0 0
\(59\) −2.79289 − 4.83743i −0.363604 − 0.629780i 0.624947 − 0.780667i \(-0.285120\pi\)
−0.988551 + 0.150887i \(0.951787\pi\)
\(60\) 0 0
\(61\) −5.91421 + 10.2437i −0.757237 + 1.31157i 0.187017 + 0.982357i \(0.440118\pi\)
−0.944254 + 0.329217i \(0.893215\pi\)
\(62\) 0 0
\(63\) 7.41421 − 1.01461i 0.934103 − 0.127829i
\(64\) 0 0
\(65\) −1.12132 + 1.94218i −0.139083 + 0.240898i
\(66\) 0 0
\(67\) 1.37868 + 2.38794i 0.168433 + 0.291734i 0.937869 − 0.346990i \(-0.112796\pi\)
−0.769436 + 0.638723i \(0.779463\pi\)
\(68\) 0 0
\(69\) 15.0711 1.81434
\(70\) 0 0
\(71\) 11.0711 1.31389 0.656947 − 0.753937i \(-0.271848\pi\)
0.656947 + 0.753937i \(0.271848\pi\)
\(72\) 0 0
\(73\) 4.70711 + 8.15295i 0.550925 + 0.954230i 0.998208 + 0.0598379i \(0.0190584\pi\)
−0.447283 + 0.894393i \(0.647608\pi\)
\(74\) 0 0
\(75\) 5.62132 − 9.73641i 0.649094 − 1.12426i
\(76\) 0 0
\(77\) 1.00000 − 2.44949i 0.113961 − 0.279145i
\(78\) 0 0
\(79\) −6.62132 + 11.4685i −0.744957 + 1.29030i 0.205258 + 0.978708i \(0.434197\pi\)
−0.950215 + 0.311595i \(0.899137\pi\)
\(80\) 0 0
\(81\) 4.74264 + 8.21449i 0.526960 + 0.912722i
\(82\) 0 0
\(83\) 12.1421 1.33277 0.666386 − 0.745607i \(-0.267840\pi\)
0.666386 + 0.745607i \(0.267840\pi\)
\(84\) 0 0
\(85\) −2.14214 −0.232347
\(86\) 0 0
\(87\) −3.20711 − 5.55487i −0.343838 − 0.595545i
\(88\) 0 0
\(89\) −6.24264 + 10.8126i −0.661719 + 1.14613i 0.318445 + 0.947941i \(0.396839\pi\)
−0.980164 + 0.198189i \(0.936494\pi\)
\(90\) 0 0
\(91\) 6.20711 + 8.00436i 0.650682 + 0.839085i
\(92\) 0 0
\(93\) −4.82843 + 8.36308i −0.500685 + 0.867211i
\(94\) 0 0
\(95\) 0.171573 + 0.297173i 0.0176030 + 0.0304893i
\(96\) 0 0
\(97\) −3.82843 −0.388718 −0.194359 − 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(98\) 0 0
\(99\) −2.82843 −0.284268
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 1232.2.q.f.177.1 4
4.3 odd 2 154.2.e.e.23.2 ✓ 4
7.2 even 3 8624.2.a.cc.1.2 2
7.4 even 3 inner 1232.2.q.f.529.1 4
7.5 odd 6 8624.2.a.bh.1.1 2
12.11 even 2 1386.2.k.t.793.2 4
28.3 even 6 1078.2.e.m.67.1 4
28.11 odd 6 154.2.e.e.67.2 yes 4
28.19 even 6 1078.2.a.x.1.2 2
28.23 odd 6 1078.2.a.t.1.1 2
28.27 even 2 1078.2.e.m.177.1 4
84.11 even 6 1386.2.k.t.991.2 4
84.23 even 6 9702.2.a.cx.1.1 2
84.47 odd 6 9702.2.a.ch.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.2 ✓ 4 4.3 odd 2
154.2.e.e.67.2 yes 4 28.11 odd 6
1078.2.a.t.1.1 2 28.23 odd 6
1078.2.a.x.1.2 2 28.19 even 6
1078.2.e.m.67.1 4 28.3 even 6
1078.2.e.m.177.1 4 28.27 even 2
1232.2.q.f.177.1 4 1.1 even 1 trivial
1232.2.q.f.529.1 4 7.4 even 3 inner
1386.2.k.t.793.2 4 12.11 even 2
1386.2.k.t.991.2 4 84.11 even 6
8624.2.a.bh.1.1 2 7.5 odd 6
8624.2.a.cc.1.2 2 7.2 even 3
9702.2.a.ch.1.2 2 84.47 odd 6
9702.2.a.cx.1.1 2 84.23 even 6