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Properties

Label 351.2.l.a.199.1
Level $351$
Weight $2$
Character 351.199
Analytic conductor $2.803$
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] = [351,2,Mod(127,351)] 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("351.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.80274911095\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 199.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 351.199
Dual form 351.2.l.a.127.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{2} -1.00000 q^{4} +(-1.50000 - 0.866025i) q^{5} +(1.50000 + 0.866025i) q^{7} +1.73205i q^{8} +(1.50000 - 2.59808i) q^{10} +3.46410i q^{11} +(-1.00000 + 3.46410i) q^{13} +(-1.50000 + 2.59808i) q^{14} -5.00000 q^{16} +(1.50000 + 2.59808i) q^{17} +(-1.50000 + 0.866025i) q^{19} +(1.50000 + 0.866025i) q^{20} -6.00000 q^{22} +(-1.50000 - 2.59808i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(-6.00000 - 1.73205i) q^{26} +(-1.50000 - 0.866025i) q^{28} +6.00000 q^{29} +(7.50000 + 4.33013i) q^{31} -5.19615i q^{32} +(-4.50000 + 2.59808i) q^{34} +(-1.50000 - 2.59808i) q^{35} +(-4.50000 - 2.59808i) q^{37} +(-1.50000 - 2.59808i) q^{38} +(1.50000 - 2.59808i) q^{40} +(10.5000 - 6.06218i) q^{41} +(0.500000 - 0.866025i) q^{43} -3.46410i q^{44} +(4.50000 - 2.59808i) q^{46} +(-4.50000 + 2.59808i) q^{47} +(-2.00000 - 3.46410i) q^{49} +(3.00000 - 1.73205i) q^{50} +(1.00000 - 3.46410i) q^{52} -6.00000 q^{53} +(3.00000 - 5.19615i) q^{55} +(-1.50000 + 2.59808i) q^{56} +10.3923i q^{58} +3.46410i q^{59} +(2.50000 - 4.33013i) q^{61} +(-7.50000 + 12.9904i) q^{62} -1.00000 q^{64} +(4.50000 - 4.33013i) q^{65} +(10.5000 - 6.06218i) q^{67} +(-1.50000 - 2.59808i) q^{68} +(4.50000 - 2.59808i) q^{70} +(7.50000 - 4.33013i) q^{71} +6.92820i q^{73} +(4.50000 - 7.79423i) q^{74} +(1.50000 - 0.866025i) q^{76} +(-3.00000 + 5.19615i) q^{77} +(5.50000 + 9.52628i) q^{79} +(7.50000 + 4.33013i) q^{80} +(10.5000 + 18.1865i) q^{82} +(-4.50000 + 2.59808i) q^{83} -5.19615i q^{85} +(1.50000 + 0.866025i) q^{86} -6.00000 q^{88} +(-13.5000 - 7.79423i) q^{89} +(-4.50000 + 4.33013i) q^{91} +(1.50000 + 2.59808i) q^{92} +(-4.50000 - 7.79423i) q^{94} +3.00000 q^{95} +(13.5000 + 7.79423i) q^{97} +(6.00000 - 3.46410i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{10} - 2 q^{13} - 3 q^{14} - 10 q^{16} + 3 q^{17} - 3 q^{19} + 3 q^{20} - 12 q^{22} - 3 q^{23} - 2 q^{25} - 12 q^{26} - 3 q^{28} + 12 q^{29} + 15 q^{31} - 9 q^{34}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/351\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(326\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 1.22474i 0.790569 + 0.612372i \(0.209785\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.50000 − 0.866025i −0.670820 − 0.387298i 0.125567 − 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) 1.50000 + 0.866025i 0.566947 + 0.327327i 0.755929 − 0.654654i \(-0.227186\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.73205i 0.612372i
\(9\) 0 0
\(10\) 1.50000 − 2.59808i 0.474342 − 0.821584i
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) −1.00000 + 3.46410i −0.277350 + 0.960769i
\(14\) −1.50000 + 2.59808i −0.400892 + 0.694365i
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 − 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) 0 0
\(19\) −1.50000 + 0.866025i −0.344124 + 0.198680i −0.662094 − 0.749421i \(-0.730332\pi\)
0.317970 + 0.948101i \(0.396999\pi\)
\(20\) 1.50000 + 0.866025i 0.335410 + 0.193649i
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −1.50000 − 2.59808i −0.312772 − 0.541736i 0.666190 − 0.745782i \(-0.267924\pi\)
−0.978961 + 0.204046i \(0.934591\pi\)
\(24\) 0 0
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) −6.00000 − 1.73205i −1.17670 − 0.339683i
\(27\) 0 0
\(28\) −1.50000 − 0.866025i −0.283473 − 0.163663i
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 7.50000 + 4.33013i 1.34704 + 0.777714i 0.987829 − 0.155543i \(-0.0497126\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 5.19615i − 0.918559i
\(33\) 0 0
\(34\) −4.50000 + 2.59808i −0.771744 + 0.445566i
\(35\) −1.50000 − 2.59808i −0.253546 − 0.439155i
\(36\) 0 0
\(37\) −4.50000 − 2.59808i −0.739795 − 0.427121i 0.0821995 − 0.996616i \(-0.473806\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −1.50000 − 2.59808i −0.243332 − 0.421464i
\(39\) 0 0
\(40\) 1.50000 − 2.59808i 0.237171 − 0.410792i
\(41\) 10.5000 − 6.06218i 1.63982 − 0.946753i 0.658932 − 0.752202i \(-0.271008\pi\)
0.980892 − 0.194551i \(-0.0623249\pi\)
\(42\) 0 0
\(43\) 0.500000 − 0.866025i 0.0762493 − 0.132068i −0.825380 − 0.564578i \(-0.809039\pi\)
0.901629 + 0.432511i \(0.142372\pi\)
\(44\) − 3.46410i − 0.522233i
\(45\) 0 0
\(46\) 4.50000 − 2.59808i 0.663489 − 0.383065i
\(47\) −4.50000 + 2.59808i −0.656392 + 0.378968i −0.790901 − 0.611944i \(-0.790388\pi\)
0.134509 + 0.990912i \(0.457054\pi\)
\(48\) 0 0
\(49\) −2.00000 − 3.46410i −0.285714 − 0.494872i
\(50\) 3.00000 − 1.73205i 0.424264 − 0.244949i
\(51\) 0 0
\(52\) 1.00000 − 3.46410i 0.138675 − 0.480384i
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 − 5.19615i 0.404520 − 0.700649i
\(56\) −1.50000 + 2.59808i −0.200446 + 0.347183i
\(57\) 0 0
\(58\) 10.3923i 1.36458i
\(59\) 3.46410i 0.450988i 0.974245 + 0.225494i \(0.0723995\pi\)
−0.974245 + 0.225494i \(0.927600\pi\)
\(60\) 0 0
\(61\) 2.50000 − 4.33013i 0.320092 − 0.554416i −0.660415 − 0.750901i \(-0.729619\pi\)
0.980507 + 0.196485i \(0.0629528\pi\)
\(62\) −7.50000 + 12.9904i −0.952501 + 1.64978i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 4.50000 − 4.33013i 0.558156 − 0.537086i
\(66\) 0 0
\(67\) 10.5000 − 6.06218i 1.28278 − 0.740613i 0.305424 − 0.952217i \(-0.401202\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) −1.50000 − 2.59808i −0.181902 − 0.315063i
\(69\) 0 0
\(70\) 4.50000 − 2.59808i 0.537853 − 0.310530i
\(71\) 7.50000 − 4.33013i 0.890086 − 0.513892i 0.0161155 − 0.999870i \(-0.494870\pi\)
0.873971 + 0.485979i \(0.161537\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 4.50000 − 7.79423i 0.523114 − 0.906061i
\(75\) 0 0
\(76\) 1.50000 − 0.866025i 0.172062 − 0.0993399i
\(77\) −3.00000 + 5.19615i −0.341882 + 0.592157i
\(78\) 0 0
\(79\) 5.50000 + 9.52628i 0.618798 + 1.07179i 0.989705 + 0.143120i \(0.0457135\pi\)
−0.370907 + 0.928670i \(0.620953\pi\)
\(80\) 7.50000 + 4.33013i 0.838525 + 0.484123i
\(81\) 0 0
\(82\) 10.5000 + 18.1865i 1.15953 + 2.00837i
\(83\) −4.50000 + 2.59808i −0.493939 + 0.285176i −0.726207 − 0.687476i \(-0.758719\pi\)
0.232268 + 0.972652i \(0.425385\pi\)
\(84\) 0 0
\(85\) − 5.19615i − 0.563602i
\(86\) 1.50000 + 0.866025i 0.161749 + 0.0933859i
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) −13.5000 − 7.79423i −1.43100 − 0.826187i −0.433800 − 0.901009i \(-0.642828\pi\)
−0.997197 + 0.0748225i \(0.976161\pi\)
\(90\) 0 0
\(91\) −4.50000 + 4.33013i −0.471728 + 0.453921i
\(92\) 1.50000 + 2.59808i 0.156386 + 0.270868i
\(93\) 0 0
\(94\) −4.50000 − 7.79423i −0.464140 − 0.803913i
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) 13.5000 + 7.79423i 1.37072 + 0.791384i 0.991018 − 0.133726i \(-0.0426942\pi\)
0.379699 + 0.925110i \(0.376028\pi\)
\(98\) 6.00000 − 3.46410i 0.606092 − 0.349927i
\(99\) 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 351.2.l.a.199.1 2
3.2 odd 2 117.2.l.a.4.1 ✓ 2
9.2 odd 6 117.2.r.a.43.1 yes 2
9.7 even 3 351.2.r.a.316.1 2
13.10 even 6 351.2.r.a.10.1 2
39.23 odd 6 117.2.r.a.49.1 yes 2
117.88 even 6 inner 351.2.l.a.127.1 2
117.101 odd 6 117.2.l.a.88.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.l.a.4.1 ✓ 2 3.2 odd 2
117.2.l.a.88.1 yes 2 117.101 odd 6
117.2.r.a.43.1 yes 2 9.2 odd 6
117.2.r.a.49.1 yes 2 39.23 odd 6
351.2.l.a.127.1 2 117.88 even 6 inner
351.2.l.a.199.1 2 1.1 even 1 trivial
351.2.r.a.10.1 2 13.10 even 6
351.2.r.a.316.1 2 9.7 even 3