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Properties

Label 230.2.e.c.137.3
Level $230$
Weight $2$
Character 230.137
Analytic conductor $1.837$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [230,2,Mod(137,230)] 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("230.137"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,16,0,0] 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: no
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 137.3
Root \(-0.382683 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 230.137
Dual form 230.2.e.c.183.3

$q$-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+(0.707107 + 0.707107i) q^{2} +(2.00000 - 2.00000i) q^{3} +1.00000i q^{4} +(-1.46508 + 1.68925i) q^{5} +2.82843 q^{6} +(2.61313 - 2.61313i) q^{7} +(-0.707107 + 0.707107i) q^{8} -5.00000i q^{9} +(-2.23044 + 0.158513i) q^{10} -0.317025i q^{11} +(2.00000 + 2.00000i) q^{12} +(-3.41421 + 3.41421i) q^{13} +3.69552 q^{14} +(0.448342 + 6.30864i) q^{15} -1.00000 q^{16} +(-3.69552 + 3.69552i) q^{17} +(3.53553 - 3.53553i) q^{18} +5.09494 q^{19} +(-1.68925 - 1.46508i) q^{20} -10.4525i q^{21} +(0.224171 - 0.224171i) q^{22} +(-3.67139 - 3.08560i) q^{23} +2.82843i q^{24} +(-0.707107 - 4.94975i) q^{25} -4.82843 q^{26} +(-4.00000 - 4.00000i) q^{27} +(2.61313 + 2.61313i) q^{28} +3.65685i q^{29} +(-4.14386 + 4.77791i) q^{30} +2.24264 q^{31} +(-0.707107 - 0.707107i) q^{32} +(-0.634051 - 0.634051i) q^{33} -5.22625 q^{34} +(0.585786 + 8.24264i) q^{35} +5.00000 q^{36} +(-2.07193 + 2.07193i) q^{37} +(3.60266 + 3.60266i) q^{38} +13.6569i q^{39} +(-0.158513 - 2.23044i) q^{40} -9.89949 q^{41} +(7.39104 - 7.39104i) q^{42} +(-5.76745 - 5.76745i) q^{43} +0.317025 q^{44} +(8.44623 + 7.32538i) q^{45} +(-0.414214 - 4.77791i) q^{46} +(1.58579 + 1.58579i) q^{47} +(-2.00000 + 2.00000i) q^{48} -6.65685i q^{49} +(3.00000 - 4.00000i) q^{50} +14.7821i q^{51} +(-3.41421 - 3.41421i) q^{52} +(5.00208 + 5.00208i) q^{53} -5.65685i q^{54} +(0.535534 + 0.464466i) q^{55} +3.69552i q^{56} +(10.1899 - 10.1899i) q^{57} +(-2.58579 + 2.58579i) q^{58} -9.65685i q^{59} +(-6.30864 + 0.448342i) q^{60} +11.8519i q^{61} +(1.58579 + 1.58579i) q^{62} +(-13.0656 - 13.0656i) q^{63} -1.00000i q^{64} +(-0.765367 - 10.7695i) q^{65} -0.896683i q^{66} +(2.38896 - 2.38896i) q^{67} +(-3.69552 - 3.69552i) q^{68} +(-13.5140 + 1.17157i) q^{69} +(-5.41421 + 6.24264i) q^{70} -4.82843 q^{71} +(3.53553 + 3.53553i) q^{72} +(6.82843 - 6.82843i) q^{73} -2.93015 q^{74} +(-11.3137 - 8.48528i) q^{75} +5.09494i q^{76} +(-0.828427 - 0.828427i) q^{77} +(-9.65685 + 9.65685i) q^{78} +4.59220 q^{79} +(1.46508 - 1.68925i) q^{80} -1.00000 q^{81} +(-7.00000 - 7.00000i) q^{82} +(-1.94061 - 1.94061i) q^{83} +10.4525 q^{84} +(-0.828427 - 11.6569i) q^{85} -8.15640i q^{86} +(7.31371 + 7.31371i) q^{87} +(0.224171 + 0.224171i) q^{88} +14.7821 q^{89} +(0.792563 + 11.1522i) q^{90} +17.8435i q^{91} +(3.08560 - 3.67139i) q^{92} +(4.48528 - 4.48528i) q^{93} +2.24264i q^{94} +(-7.46447 + 8.60660i) q^{95} -2.82843 q^{96} +(3.06147 - 3.06147i) q^{97} +(4.70711 - 4.70711i) q^{98} -1.58513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{3} + 16 q^{12} - 16 q^{13} - 8 q^{16} - 8 q^{23} - 16 q^{26} - 32 q^{27} - 16 q^{31} + 16 q^{35} + 40 q^{36} + 8 q^{46} + 24 q^{47} - 16 q^{48} + 24 q^{50} - 16 q^{52} - 24 q^{55} - 32 q^{58}+ \cdots + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 + 0.707107i 0.500000 + 0.500000i
\(3\) 2.00000 − 2.00000i 1.15470 − 1.15470i 0.169102 − 0.985599i \(-0.445913\pi\)
0.985599 − 0.169102i \(-0.0540867\pi\)
\(4\) 1.00000i 0.500000i
\(5\) −1.46508 + 1.68925i −0.655202 + 0.755454i
\(6\) 2.82843 1.15470
\(7\) 2.61313 − 2.61313i 0.987669 − 0.987669i −0.0122561 − 0.999925i \(-0.503901\pi\)
0.999925 + 0.0122561i \(0.00390134\pi\)
\(8\) −0.707107 + 0.707107i −0.250000 + 0.250000i
\(9\) − 5.00000i − 1.66667i
\(10\) −2.23044 + 0.158513i −0.705328 + 0.0501261i
\(11\) − 0.317025i − 0.0955867i −0.998857 − 0.0477934i \(-0.984781\pi\)
0.998857 − 0.0477934i \(-0.0152189\pi\)
\(12\) 2.00000 + 2.00000i 0.577350 + 0.577350i
\(13\) −3.41421 + 3.41421i −0.946932 + 0.946932i −0.998661 − 0.0517287i \(-0.983527\pi\)
0.0517287 + 0.998661i \(0.483527\pi\)
\(14\) 3.69552 0.987669
\(15\) 0.448342 + 6.30864i 0.115761 + 1.62888i
\(16\) −1.00000 −0.250000
\(17\) −3.69552 + 3.69552i −0.896295 + 0.896295i −0.995106 − 0.0988114i \(-0.968496\pi\)
0.0988114 + 0.995106i \(0.468496\pi\)
\(18\) 3.53553 − 3.53553i 0.833333 − 0.833333i
\(19\) 5.09494 1.16886 0.584429 − 0.811445i \(-0.301318\pi\)
0.584429 + 0.811445i \(0.301318\pi\)
\(20\) −1.68925 − 1.46508i −0.377727 − 0.327601i
\(21\) − 10.4525i − 2.28092i
\(22\) 0.224171 − 0.224171i 0.0477934 − 0.0477934i
\(23\) −3.67139 − 3.08560i −0.765537 − 0.643392i
\(24\) 2.82843i 0.577350i
\(25\) −0.707107 − 4.94975i −0.141421 − 0.989949i
\(26\) −4.82843 −0.946932
\(27\) −4.00000 − 4.00000i −0.769800 − 0.769800i
\(28\) 2.61313 + 2.61313i 0.493834 + 0.493834i
\(29\) 3.65685i 0.679061i 0.940595 + 0.339530i \(0.110268\pi\)
−0.940595 + 0.339530i \(0.889732\pi\)
\(30\) −4.14386 + 4.77791i −0.756562 + 0.872323i
\(31\) 2.24264 0.402790 0.201395 − 0.979510i \(-0.435452\pi\)
0.201395 + 0.979510i \(0.435452\pi\)
\(32\) −0.707107 − 0.707107i −0.125000 − 0.125000i
\(33\) −0.634051 − 0.634051i −0.110374 − 0.110374i
\(34\) −5.22625 −0.896295
\(35\) 0.585786 + 8.24264i 0.0990160 + 1.39326i
\(36\) 5.00000 0.833333
\(37\) −2.07193 + 2.07193i −0.340623 + 0.340623i −0.856602 − 0.515978i \(-0.827428\pi\)
0.515978 + 0.856602i \(0.327428\pi\)
\(38\) 3.60266 + 3.60266i 0.584429 + 0.584429i
\(39\) 13.6569i 2.18685i
\(40\) −0.158513 − 2.23044i −0.0250631 − 0.352664i
\(41\) −9.89949 −1.54604 −0.773021 − 0.634381i \(-0.781255\pi\)
−0.773021 + 0.634381i \(0.781255\pi\)
\(42\) 7.39104 − 7.39104i 1.14046 − 1.14046i
\(43\) −5.76745 − 5.76745i −0.879528 − 0.879528i 0.113958 − 0.993486i \(-0.463647\pi\)
−0.993486 + 0.113958i \(0.963647\pi\)
\(44\) 0.317025 0.0477934
\(45\) 8.44623 + 7.32538i 1.25909 + 1.09200i
\(46\) −0.414214 − 4.77791i −0.0610725 − 0.704464i
\(47\) 1.58579 + 1.58579i 0.231311 + 0.231311i 0.813240 − 0.581929i \(-0.197702\pi\)
−0.581929 + 0.813240i \(0.697702\pi\)
\(48\) −2.00000 + 2.00000i −0.288675 + 0.288675i
\(49\) − 6.65685i − 0.950979i
\(50\) 3.00000 − 4.00000i 0.424264 − 0.565685i
\(51\) 14.7821i 2.06990i
\(52\) −3.41421 − 3.41421i −0.473466 − 0.473466i
\(53\) 5.00208 + 5.00208i 0.687089 + 0.687089i 0.961587 − 0.274499i \(-0.0885119\pi\)
−0.274499 + 0.961587i \(0.588512\pi\)
\(54\) − 5.65685i − 0.769800i
\(55\) 0.535534 + 0.464466i 0.0722114 + 0.0626286i
\(56\) 3.69552i 0.493834i
\(57\) 10.1899 − 10.1899i 1.34968 − 1.34968i
\(58\) −2.58579 + 2.58579i −0.339530 + 0.339530i
\(59\) − 9.65685i − 1.25722i −0.777723 − 0.628608i \(-0.783625\pi\)
0.777723 − 0.628608i \(-0.216375\pi\)
\(60\) −6.30864 + 0.448342i −0.814442 + 0.0578806i
\(61\) 11.8519i 1.51748i 0.651392 + 0.758742i \(0.274185\pi\)
−0.651392 + 0.758742i \(0.725815\pi\)
\(62\) 1.58579 + 1.58579i 0.201395 + 0.201395i
\(63\) −13.0656 − 13.0656i −1.64611 − 1.64611i
\(64\) − 1.00000i − 0.125000i
\(65\) −0.765367 − 10.7695i −0.0949321 − 1.33580i
\(66\) − 0.896683i − 0.110374i
\(67\) 2.38896 − 2.38896i 0.291857 − 0.291857i −0.545956 − 0.837814i \(-0.683834\pi\)
0.837814 + 0.545956i \(0.183834\pi\)
\(68\) −3.69552 − 3.69552i −0.448147 − 0.448147i
\(69\) −13.5140 + 1.17157i −1.62689 + 0.141041i
\(70\) −5.41421 + 6.24264i −0.647122 + 0.746138i
\(71\) −4.82843 −0.573029 −0.286514 − 0.958076i \(-0.592497\pi\)
−0.286514 + 0.958076i \(0.592497\pi\)
\(72\) 3.53553 + 3.53553i 0.416667 + 0.416667i
\(73\) 6.82843 − 6.82843i 0.799207 − 0.799207i −0.183764 − 0.982970i \(-0.558828\pi\)
0.982970 + 0.183764i \(0.0588281\pi\)
\(74\) −2.93015 −0.340623
\(75\) −11.3137 − 8.48528i −1.30639 − 0.979796i
\(76\) 5.09494i 0.584429i
\(77\) −0.828427 − 0.828427i −0.0944080 − 0.0944080i
\(78\) −9.65685 + 9.65685i −1.09342 + 1.09342i
\(79\) 4.59220 0.516663 0.258331 − 0.966056i \(-0.416827\pi\)
0.258331 + 0.966056i \(0.416827\pi\)
\(80\) 1.46508 − 1.68925i 0.163800 − 0.188863i
\(81\) −1.00000 −0.111111
\(82\) −7.00000 − 7.00000i −0.773021 − 0.773021i
\(83\) −1.94061 − 1.94061i −0.213010 − 0.213010i 0.592535 − 0.805545i \(-0.298127\pi\)
−0.805545 + 0.592535i \(0.798127\pi\)
\(84\) 10.4525 1.14046
\(85\) −0.828427 − 11.6569i −0.0898555 − 1.26436i
\(86\) − 8.15640i − 0.879528i
\(87\) 7.31371 + 7.31371i 0.784112 + 0.784112i
\(88\) 0.224171 + 0.224171i 0.0238967 + 0.0238967i
\(89\) 14.7821 1.56690 0.783448 − 0.621457i \(-0.213459\pi\)
0.783448 + 0.621457i \(0.213459\pi\)
\(90\) 0.792563 + 11.1522i 0.0835435 + 1.17555i
\(91\) 17.8435i 1.87051i
\(92\) 3.08560 − 3.67139i 0.321696 − 0.382768i
\(93\) 4.48528 − 4.48528i 0.465102 − 0.465102i
\(94\) 2.24264i 0.231311i
\(95\) −7.46447 + 8.60660i −0.765838 + 0.883019i
\(96\) −2.82843 −0.288675
\(97\) 3.06147 − 3.06147i 0.310845 − 0.310845i −0.534392 − 0.845237i \(-0.679459\pi\)
0.845237 + 0.534392i \(0.179459\pi\)
\(98\) 4.70711 − 4.70711i 0.475490 − 0.475490i
\(99\) −1.58513 −0.159311
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 230.2.e.c.137.3 ✓ 8
5.2 odd 4 1150.2.e.a.643.2 8
5.3 odd 4 inner 230.2.e.c.183.4 yes 8
5.4 even 2 1150.2.e.a.1057.1 8
23.22 odd 2 inner 230.2.e.c.137.4 yes 8
115.22 even 4 1150.2.e.a.643.1 8
115.68 even 4 inner 230.2.e.c.183.3 yes 8
115.114 odd 2 1150.2.e.a.1057.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.e.c.137.3 ✓ 8 1.1 even 1 trivial
230.2.e.c.137.4 yes 8 23.22 odd 2 inner
230.2.e.c.183.3 yes 8 115.68 even 4 inner
230.2.e.c.183.4 yes 8 5.3 odd 4 inner
1150.2.e.a.643.1 8 115.22 even 4
1150.2.e.a.643.2 8 5.2 odd 4
1150.2.e.a.1057.1 8 5.4 even 2
1150.2.e.a.1057.2 8 115.114 odd 2