Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [154,2,Mod(23,154)] 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("154.23"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(154, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 154.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.22969619113\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 23.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 154.23
Dual form 154.2.e.e.67.2

$q$-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+(-0.500000 + 0.866025i) q^{2} +(1.20711 + 2.09077i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.292893 - 0.507306i) q^{5} -2.41421 q^{6} +(1.62132 + 2.09077i) q^{7} +1.00000 q^{8} +(-1.41421 + 2.44949i) q^{9} +(0.292893 + 0.507306i) q^{10} +(-0.500000 - 0.866025i) q^{11} +(1.20711 - 2.09077i) q^{12} -3.82843 q^{13} +(-2.62132 + 0.358719i) q^{14} +1.41421 q^{15} +(-0.500000 + 0.866025i) q^{16} +(-1.82843 - 3.16693i) q^{17} +(-1.41421 - 2.44949i) q^{18} +(0.292893 - 0.507306i) q^{19} -0.585786 q^{20} +(-2.41421 + 5.91359i) q^{21} +1.00000 q^{22} +(3.12132 - 5.40629i) q^{23} +(1.20711 + 2.09077i) q^{24} +(2.32843 + 4.03295i) q^{25} +(1.91421 - 3.31552i) q^{26} +0.414214 q^{27} +(1.00000 - 2.44949i) q^{28} +2.65685 q^{29} +(-0.707107 + 1.22474i) q^{30} +(2.00000 + 3.46410i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(1.20711 - 2.09077i) q^{33} +3.65685 q^{34} +(1.53553 - 0.210133i) q^{35} +2.82843 q^{36} +(4.70711 - 8.15295i) q^{37} +(0.292893 + 0.507306i) q^{38} +(-4.62132 - 8.00436i) q^{39} +(0.292893 - 0.507306i) q^{40} -5.41421 q^{41} +(-3.91421 - 5.04757i) q^{42} -5.65685 q^{43} +(-0.500000 + 0.866025i) q^{44} +(0.828427 + 1.43488i) q^{45} +(3.12132 + 5.40629i) q^{46} +(5.24264 - 9.08052i) q^{47} -2.41421 q^{48} +(-1.74264 + 6.77962i) q^{49} -4.65685 q^{50} +(4.41421 - 7.64564i) q^{51} +(1.91421 + 3.31552i) q^{52} +(-3.94975 - 6.84116i) q^{53} +(-0.207107 + 0.358719i) q^{54} -0.585786 q^{55} +(1.62132 + 2.09077i) q^{56} +1.41421 q^{57} +(-1.32843 + 2.30090i) q^{58} +(2.79289 + 4.83743i) q^{59} +(-0.707107 - 1.22474i) q^{60} +(-5.91421 + 10.2437i) q^{61} -4.00000 q^{62} +(-7.41421 + 1.01461i) q^{63} +1.00000 q^{64} +(-1.12132 + 1.94218i) q^{65} +(1.20711 + 2.09077i) q^{66} +(-1.37868 - 2.38794i) q^{67} +(-1.82843 + 3.16693i) q^{68} +15.0711 q^{69} +(-0.585786 + 1.43488i) q^{70} -11.0711 q^{71} +(-1.41421 + 2.44949i) q^{72} +(4.70711 + 8.15295i) q^{73} +(4.70711 + 8.15295i) q^{74} +(-5.62132 + 9.73641i) q^{75} -0.585786 q^{76} +(1.00000 - 2.44949i) q^{77} +9.24264 q^{78} +(6.62132 - 11.4685i) q^{79} +(0.292893 + 0.507306i) q^{80} +(4.74264 + 8.21449i) q^{81} +(2.70711 - 4.68885i) q^{82} -12.1421 q^{83} +(6.32843 - 0.866025i) q^{84} -2.14214 q^{85} +(2.82843 - 4.89898i) q^{86} +(3.20711 + 5.55487i) q^{87} +(-0.500000 - 0.866025i) q^{88} +(-6.24264 + 10.8126i) q^{89} -1.65685 q^{90} +(-6.20711 - 8.00436i) q^{91} -6.24264 q^{92} +(-4.82843 + 8.36308i) q^{93} +(5.24264 + 9.08052i) q^{94} +(-0.171573 - 0.297173i) q^{95} +(1.20711 - 2.09077i) q^{96} -3.82843 q^{97} +(-5.00000 - 4.89898i) q^{98} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} + 4 q^{8} + 4 q^{10} - 2 q^{11} + 2 q^{12} - 4 q^{13} - 2 q^{14} - 2 q^{16} + 4 q^{17} + 4 q^{19} - 8 q^{20} - 4 q^{21} + 4 q^{22} + 4 q^{23}+ \cdots - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.353553 + 0.612372i
\(3\) 1.20711 + 2.09077i 0.696923 + 1.20711i 0.969528 + 0.244981i \(0.0787816\pi\)
−0.272605 + 0.962126i \(0.587885\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0.292893 − 0.507306i 0.130986 − 0.226874i −0.793071 − 0.609129i \(-0.791519\pi\)
0.924057 + 0.382255i \(0.124852\pi\)
\(6\) −2.41421 −0.985599
\(7\) 1.62132 + 2.09077i 0.612801 + 0.790237i
\(8\) 1.00000 0.353553
\(9\) −1.41421 + 2.44949i −0.471405 + 0.816497i
\(10\) 0.292893 + 0.507306i 0.0926210 + 0.160424i
\(11\) −0.500000 − 0.866025i −0.150756 − 0.261116i
\(12\) 1.20711 − 2.09077i 0.348462 − 0.603553i
\(13\) −3.82843 −1.06181 −0.530907 − 0.847430i \(-0.678149\pi\)
−0.530907 + 0.847430i \(0.678149\pi\)
\(14\) −2.62132 + 0.358719i −0.700577 + 0.0958718i
\(15\) 1.41421 0.365148
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −1.82843 − 3.16693i −0.443459 − 0.768093i 0.554485 − 0.832194i \(-0.312915\pi\)
−0.997943 + 0.0641009i \(0.979582\pi\)
\(18\) −1.41421 − 2.44949i −0.333333 − 0.577350i
\(19\) 0.292893 − 0.507306i 0.0671943 − 0.116384i −0.830471 − 0.557062i \(-0.811929\pi\)
0.897665 + 0.440678i \(0.145262\pi\)
\(20\) −0.585786 −0.130986
\(21\) −2.41421 + 5.91359i −0.526825 + 1.29045i
\(22\) 1.00000 0.213201
\(23\) 3.12132 − 5.40629i 0.650840 − 1.12729i −0.332079 − 0.943252i \(-0.607750\pi\)
0.982919 − 0.184037i \(-0.0589166\pi\)
\(24\) 1.20711 + 2.09077i 0.246400 + 0.426777i
\(25\) 2.32843 + 4.03295i 0.465685 + 0.806591i
\(26\) 1.91421 − 3.31552i 0.375408 − 0.650226i
\(27\) 0.414214 0.0797154
\(28\) 1.00000 − 2.44949i 0.188982 − 0.462910i
\(29\) 2.65685 0.493365 0.246683 − 0.969096i \(-0.420659\pi\)
0.246683 + 0.969096i \(0.420659\pi\)
\(30\) −0.707107 + 1.22474i −0.129099 + 0.223607i
\(31\) 2.00000 + 3.46410i 0.359211 + 0.622171i 0.987829 − 0.155543i \(-0.0497126\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 1.20711 − 2.09077i 0.210130 − 0.363956i
\(34\) 3.65685 0.627145
\(35\) 1.53553 − 0.210133i 0.259553 − 0.0355190i
\(36\) 2.82843 0.471405
\(37\) 4.70711 − 8.15295i 0.773844 − 1.34034i −0.161599 − 0.986857i \(-0.551665\pi\)
0.935442 − 0.353480i \(-0.115002\pi\)
\(38\) 0.292893 + 0.507306i 0.0475136 + 0.0822959i
\(39\) −4.62132 − 8.00436i −0.740003 − 1.28172i
\(40\) 0.292893 − 0.507306i 0.0463105 − 0.0802121i
\(41\) −5.41421 −0.845558 −0.422779 − 0.906233i \(-0.638945\pi\)
−0.422779 + 0.906233i \(0.638945\pi\)
\(42\) −3.91421 − 5.04757i −0.603976 − 0.778856i
\(43\) −5.65685 −0.862662 −0.431331 − 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) −0.500000 + 0.866025i −0.0753778 + 0.130558i
\(45\) 0.828427 + 1.43488i 0.123495 + 0.213899i
\(46\) 3.12132 + 5.40629i 0.460214 + 0.797113i
\(47\) 5.24264 − 9.08052i 0.764718 − 1.32453i −0.175678 − 0.984448i \(-0.556212\pi\)
0.940396 − 0.340082i \(-0.110455\pi\)
\(48\) −2.41421 −0.348462
\(49\) −1.74264 + 6.77962i −0.248949 + 0.968517i
\(50\) −4.65685 −0.658579
\(51\) 4.41421 − 7.64564i 0.618114 − 1.07060i
\(52\) 1.91421 + 3.31552i 0.265454 + 0.459779i
\(53\) −3.94975 − 6.84116i −0.542540 − 0.939706i −0.998757 − 0.0498379i \(-0.984130\pi\)
0.456218 − 0.889868i \(-0.349204\pi\)
\(54\) −0.207107 + 0.358719i −0.0281837 + 0.0488155i
\(55\) −0.585786 −0.0789874
\(56\) 1.62132 + 2.09077i 0.216658 + 0.279391i
\(57\) 1.41421 0.187317
\(58\) −1.32843 + 2.30090i −0.174431 + 0.302123i
\(59\) 2.79289 + 4.83743i 0.363604 + 0.629780i 0.988551 − 0.150887i \(-0.0482129\pi\)
−0.624947 + 0.780667i \(0.714880\pi\)
\(60\) −0.707107 − 1.22474i −0.0912871 − 0.158114i
\(61\) −5.91421 + 10.2437i −0.757237 + 1.31157i 0.187017 + 0.982357i \(0.440118\pi\)
−0.944254 + 0.329217i \(0.893215\pi\)
\(62\) −4.00000 −0.508001
\(63\) −7.41421 + 1.01461i −0.934103 + 0.127829i
\(64\) 1.00000 0.125000
\(65\) −1.12132 + 1.94218i −0.139083 + 0.240898i
\(66\) 1.20711 + 2.09077i 0.148585 + 0.257356i
\(67\) −1.37868 − 2.38794i −0.168433 − 0.291734i 0.769436 − 0.638723i \(-0.220537\pi\)
−0.937869 + 0.346990i \(0.887204\pi\)
\(68\) −1.82843 + 3.16693i −0.221729 + 0.384047i
\(69\) 15.0711 1.81434
\(70\) −0.585786 + 1.43488i −0.0700149 + 0.171501i
\(71\) −11.0711 −1.31389 −0.656947 − 0.753937i \(-0.728152\pi\)
−0.656947 + 0.753937i \(0.728152\pi\)
\(72\) −1.41421 + 2.44949i −0.166667 + 0.288675i
\(73\) 4.70711 + 8.15295i 0.550925 + 0.954230i 0.998208 + 0.0598379i \(0.0190584\pi\)
−0.447283 + 0.894393i \(0.647608\pi\)
\(74\) 4.70711 + 8.15295i 0.547190 + 0.947761i
\(75\) −5.62132 + 9.73641i −0.649094 + 1.12426i
\(76\) −0.585786 −0.0671943
\(77\) 1.00000 − 2.44949i 0.113961 − 0.279145i
\(78\) 9.24264 1.04652
\(79\) 6.62132 − 11.4685i 0.744957 − 1.29030i −0.205258 − 0.978708i \(-0.565803\pi\)
0.950215 − 0.311595i \(-0.100863\pi\)
\(80\) 0.292893 + 0.507306i 0.0327465 + 0.0567185i
\(81\) 4.74264 + 8.21449i 0.526960 + 0.912722i
\(82\) 2.70711 − 4.68885i 0.298950 − 0.517796i
\(83\) −12.1421 −1.33277 −0.666386 − 0.745607i \(-0.732160\pi\)
−0.666386 + 0.745607i \(0.732160\pi\)
\(84\) 6.32843 − 0.866025i 0.690488 − 0.0944911i
\(85\) −2.14214 −0.232347
\(86\) 2.82843 − 4.89898i 0.304997 − 0.528271i
\(87\) 3.20711 + 5.55487i 0.343838 + 0.595545i
\(88\) −0.500000 − 0.866025i −0.0533002 − 0.0923186i
\(89\) −6.24264 + 10.8126i −0.661719 + 1.14613i 0.318445 + 0.947941i \(0.396839\pi\)
−0.980164 + 0.198189i \(0.936494\pi\)
\(90\) −1.65685 −0.174648
\(91\) −6.20711 − 8.00436i −0.650682 − 0.839085i
\(92\) −6.24264 −0.650840
\(93\) −4.82843 + 8.36308i −0.500685 + 0.867211i
\(94\) 5.24264 + 9.08052i 0.540737 + 0.936584i
\(95\) −0.171573 − 0.297173i −0.0176030 − 0.0304893i
\(96\) 1.20711 − 2.09077i 0.123200 − 0.213388i
\(97\) −3.82843 −0.388718 −0.194359 − 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(98\) −5.00000 − 4.89898i −0.505076 − 0.494872i
\(99\) 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 154.2.e.e.23.2 ✓ 4
3.2 odd 2 1386.2.k.t.793.2 4
4.3 odd 2 1232.2.q.f.177.1 4
7.2 even 3 1078.2.a.t.1.1 2
7.3 odd 6 1078.2.e.m.67.1 4
7.4 even 3 inner 154.2.e.e.67.2 yes 4
7.5 odd 6 1078.2.a.x.1.2 2
7.6 odd 2 1078.2.e.m.177.1 4
21.2 odd 6 9702.2.a.cx.1.1 2
21.5 even 6 9702.2.a.ch.1.2 2
21.11 odd 6 1386.2.k.t.991.2 4
28.11 odd 6 1232.2.q.f.529.1 4
28.19 even 6 8624.2.a.bh.1.1 2
28.23 odd 6 8624.2.a.cc.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.2 ✓ 4 1.1 even 1 trivial
154.2.e.e.67.2 yes 4 7.4 even 3 inner
1078.2.a.t.1.1 2 7.2 even 3
1078.2.a.x.1.2 2 7.5 odd 6
1078.2.e.m.67.1 4 7.3 odd 6
1078.2.e.m.177.1 4 7.6 odd 2
1232.2.q.f.177.1 4 4.3 odd 2
1232.2.q.f.529.1 4 28.11 odd 6
1386.2.k.t.793.2 4 3.2 odd 2
1386.2.k.t.991.2 4 21.11 odd 6
8624.2.a.bh.1.1 2 28.19 even 6
8624.2.a.cc.1.2 2 28.23 odd 6
9702.2.a.ch.1.2 2 21.5 even 6
9702.2.a.cx.1.1 2 21.2 odd 6