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Properties

Label 338.2.c.a.191.1
Level $338$
Weight $2$
Character 338.191
Analytic conductor $2.699$
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] = [338,2,Mod(191,338)] 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("338.191"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,-1] 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: \(2.69894358832\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 191.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 338.191
Dual form 338.2.c.a.315.1

$q$-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+(-0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +3.00000 q^{5} +(-0.500000 - 0.866025i) q^{6} +(-0.500000 - 0.866025i) q^{7} +1.00000 q^{8} +(1.00000 + 1.73205i) q^{9} +(-1.50000 + 2.59808i) q^{10} +(3.00000 - 5.19615i) q^{11} +1.00000 q^{12} +1.00000 q^{14} +(-1.50000 + 2.59808i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(1.50000 + 2.59808i) q^{17} -2.00000 q^{18} +(1.00000 + 1.73205i) q^{19} +(-1.50000 - 2.59808i) q^{20} +1.00000 q^{21} +(3.00000 + 5.19615i) q^{22} +(-0.500000 + 0.866025i) q^{24} +4.00000 q^{25} -5.00000 q^{27} +(-0.500000 + 0.866025i) q^{28} +(-3.00000 + 5.19615i) q^{29} +(-1.50000 - 2.59808i) q^{30} +4.00000 q^{31} +(-0.500000 - 0.866025i) q^{32} +(3.00000 + 5.19615i) q^{33} -3.00000 q^{34} +(-1.50000 - 2.59808i) q^{35} +(1.00000 - 1.73205i) q^{36} +(-3.50000 + 6.06218i) q^{37} -2.00000 q^{38} +3.00000 q^{40} +(-0.500000 + 0.866025i) q^{42} +(0.500000 + 0.866025i) q^{43} -6.00000 q^{44} +(3.00000 + 5.19615i) q^{45} -3.00000 q^{47} +(-0.500000 - 0.866025i) q^{48} +(3.00000 - 5.19615i) q^{49} +(-2.00000 + 3.46410i) q^{50} -3.00000 q^{51} +(2.50000 - 4.33013i) q^{54} +(9.00000 - 15.5885i) q^{55} +(-0.500000 - 0.866025i) q^{56} -2.00000 q^{57} +(-3.00000 - 5.19615i) q^{58} +(-3.00000 - 5.19615i) q^{59} +3.00000 q^{60} +(-4.00000 - 6.92820i) q^{61} +(-2.00000 + 3.46410i) q^{62} +(1.00000 - 1.73205i) q^{63} +1.00000 q^{64} -6.00000 q^{66} +(7.00000 - 12.1244i) q^{67} +(1.50000 - 2.59808i) q^{68} +3.00000 q^{70} +(-1.50000 - 2.59808i) q^{71} +(1.00000 + 1.73205i) q^{72} -2.00000 q^{73} +(-3.50000 - 6.06218i) q^{74} +(-2.00000 + 3.46410i) q^{75} +(1.00000 - 1.73205i) q^{76} -6.00000 q^{77} +8.00000 q^{79} +(-1.50000 + 2.59808i) q^{80} +(-0.500000 + 0.866025i) q^{81} -12.0000 q^{83} +(-0.500000 - 0.866025i) q^{84} +(4.50000 + 7.79423i) q^{85} -1.00000 q^{86} +(-3.00000 - 5.19615i) q^{87} +(3.00000 - 5.19615i) q^{88} +(-3.00000 + 5.19615i) q^{89} -6.00000 q^{90} +(-2.00000 + 3.46410i) q^{93} +(1.50000 - 2.59808i) q^{94} +(3.00000 + 5.19615i) q^{95} +1.00000 q^{96} +(-5.00000 - 8.66025i) q^{97} +(3.00000 + 5.19615i) q^{98} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} - q^{4} + 6 q^{5} - q^{6} - q^{7} + 2 q^{8} + 2 q^{9} - 3 q^{10} + 6 q^{11} + 2 q^{12} + 2 q^{14} - 3 q^{15} - q^{16} + 3 q^{17} - 4 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + 6 q^{22}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.353553 + 0.612372i
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i −0.973494 − 0.228714i \(-0.926548\pi\)
0.684819 + 0.728714i \(0.259881\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −0.500000 − 0.866025i −0.204124 − 0.353553i
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i 0.755929 − 0.654654i \(-0.227186\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 + 1.73205i 0.333333 + 0.577350i
\(10\) −1.50000 + 2.59808i −0.474342 + 0.821584i
\(11\) 3.00000 − 5.19615i 0.904534 − 1.56670i 0.0829925 − 0.996550i \(-0.473552\pi\)
0.821541 − 0.570149i \(-0.193114\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) −1.50000 + 2.59808i −0.387298 + 0.670820i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 − 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) −2.00000 −0.471405
\(19\) 1.00000 + 1.73205i 0.229416 + 0.397360i 0.957635 − 0.287984i \(-0.0929851\pi\)
−0.728219 + 0.685344i \(0.759652\pi\)
\(20\) −1.50000 − 2.59808i −0.335410 − 0.580948i
\(21\) 1.00000 0.218218
\(22\) 3.00000 + 5.19615i 0.639602 + 1.10782i
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) −0.500000 + 0.866025i −0.102062 + 0.176777i
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) −0.500000 + 0.866025i −0.0944911 + 0.163663i
\(29\) −3.00000 + 5.19615i −0.557086 + 0.964901i 0.440652 + 0.897678i \(0.354747\pi\)
−0.997738 + 0.0672232i \(0.978586\pi\)
\(30\) −1.50000 − 2.59808i −0.273861 − 0.474342i
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 3.00000 + 5.19615i 0.522233 + 0.904534i
\(34\) −3.00000 −0.514496
\(35\) −1.50000 − 2.59808i −0.253546 − 0.439155i
\(36\) 1.00000 − 1.73205i 0.166667 − 0.288675i
\(37\) −3.50000 + 6.06218i −0.575396 + 0.996616i 0.420602 + 0.907245i \(0.361819\pi\)
−0.995998 + 0.0893706i \(0.971514\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) −0.500000 + 0.866025i −0.0771517 + 0.133631i
\(43\) 0.500000 + 0.866025i 0.0762493 + 0.132068i 0.901629 − 0.432511i \(-0.142372\pi\)
−0.825380 + 0.564578i \(0.809039\pi\)
\(44\) −6.00000 −0.904534
\(45\) 3.00000 + 5.19615i 0.447214 + 0.774597i
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) −0.500000 − 0.866025i −0.0721688 − 0.125000i
\(49\) 3.00000 − 5.19615i 0.428571 − 0.742307i
\(50\) −2.00000 + 3.46410i −0.282843 + 0.489898i
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 2.50000 − 4.33013i 0.340207 − 0.589256i
\(55\) 9.00000 − 15.5885i 1.21356 − 2.10195i
\(56\) −0.500000 − 0.866025i −0.0668153 − 0.115728i
\(57\) −2.00000 −0.264906
\(58\) −3.00000 − 5.19615i −0.393919 − 0.682288i
\(59\) −3.00000 − 5.19615i −0.390567 − 0.676481i 0.601958 − 0.798528i \(-0.294388\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(60\) 3.00000 0.387298
\(61\) −4.00000 − 6.92820i −0.512148 − 0.887066i −0.999901 − 0.0140840i \(-0.995517\pi\)
0.487753 − 0.872982i \(-0.337817\pi\)
\(62\) −2.00000 + 3.46410i −0.254000 + 0.439941i
\(63\) 1.00000 − 1.73205i 0.125988 − 0.218218i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 7.00000 − 12.1244i 0.855186 − 1.48123i −0.0212861 − 0.999773i \(-0.506776\pi\)
0.876472 − 0.481452i \(-0.159891\pi\)
\(68\) 1.50000 − 2.59808i 0.181902 − 0.315063i
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) −1.50000 − 2.59808i −0.178017 − 0.308335i 0.763184 − 0.646181i \(-0.223635\pi\)
−0.941201 + 0.337846i \(0.890302\pi\)
\(72\) 1.00000 + 1.73205i 0.117851 + 0.204124i
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −3.50000 − 6.06218i −0.406867 − 0.704714i
\(75\) −2.00000 + 3.46410i −0.230940 + 0.400000i
\(76\) 1.00000 − 1.73205i 0.114708 − 0.198680i
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.50000 + 2.59808i −0.167705 + 0.290474i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −0.500000 − 0.866025i −0.0545545 − 0.0944911i
\(85\) 4.50000 + 7.79423i 0.488094 + 0.845403i
\(86\) −1.00000 −0.107833
\(87\) −3.00000 − 5.19615i −0.321634 − 0.557086i
\(88\) 3.00000 − 5.19615i 0.319801 − 0.553912i
\(89\) −3.00000 + 5.19615i −0.317999 + 0.550791i −0.980071 − 0.198650i \(-0.936344\pi\)
0.662071 + 0.749441i \(0.269678\pi\)
\(90\) −6.00000 −0.632456
\(91\) 0 0
\(92\) 0 0
\(93\) −2.00000 + 3.46410i −0.207390 + 0.359211i
\(94\) 1.50000 − 2.59808i 0.154713 − 0.267971i
\(95\) 3.00000 + 5.19615i 0.307794 + 0.533114i
\(96\) 1.00000 0.102062
\(97\) −5.00000 − 8.66025i −0.507673 − 0.879316i −0.999961 − 0.00888289i \(-0.997172\pi\)
0.492287 − 0.870433i \(-0.336161\pi\)
\(98\) 3.00000 + 5.19615i 0.303046 + 0.524891i
\(99\) 12.0000 1.20605
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 338.2.c.a.191.1 2
13.2 odd 12 338.2.e.a.23.1 4
13.3 even 3 inner 338.2.c.a.315.1 2
13.4 even 6 26.2.a.a.1.1 ✓ 1
13.5 odd 4 338.2.e.a.147.2 4
13.6 odd 12 338.2.b.c.337.1 2
13.7 odd 12 338.2.b.c.337.2 2
13.8 odd 4 338.2.e.a.147.1 4
13.9 even 3 338.2.a.f.1.1 1
13.10 even 6 338.2.c.d.315.1 2
13.11 odd 12 338.2.e.a.23.2 4
13.12 even 2 338.2.c.d.191.1 2
39.17 odd 6 234.2.a.e.1.1 1
39.20 even 12 3042.2.b.a.1351.1 2
39.32 even 12 3042.2.b.a.1351.2 2
39.35 odd 6 3042.2.a.a.1.1 1
52.7 even 12 2704.2.f.d.337.2 2
52.19 even 12 2704.2.f.d.337.1 2
52.35 odd 6 2704.2.a.f.1.1 1
52.43 odd 6 208.2.a.a.1.1 1
65.4 even 6 650.2.a.j.1.1 1
65.9 even 6 8450.2.a.c.1.1 1
65.17 odd 12 650.2.b.d.599.1 2
65.43 odd 12 650.2.b.d.599.2 2
91.4 even 6 1274.2.f.p.79.1 2
91.17 odd 6 1274.2.f.r.79.1 2
91.30 even 6 1274.2.f.p.1145.1 2
91.69 odd 6 1274.2.a.d.1.1 1
91.82 odd 6 1274.2.f.r.1145.1 2
104.43 odd 6 832.2.a.i.1.1 1
104.69 even 6 832.2.a.d.1.1 1
117.4 even 6 2106.2.e.ba.703.1 2
117.43 even 6 2106.2.e.ba.1405.1 2
117.56 odd 6 2106.2.e.b.1405.1 2
117.95 odd 6 2106.2.e.b.703.1 2
143.43 odd 6 3146.2.a.n.1.1 1
156.95 even 6 1872.2.a.q.1.1 1
195.17 even 12 5850.2.e.a.5149.2 2
195.134 odd 6 5850.2.a.p.1.1 1
195.173 even 12 5850.2.e.a.5149.1 2
208.43 odd 12 3328.2.b.j.1665.2 2
208.69 even 12 3328.2.b.m.1665.1 2
208.147 odd 12 3328.2.b.j.1665.1 2
208.173 even 12 3328.2.b.m.1665.2 2
221.186 even 6 7514.2.a.c.1.1 1
247.56 odd 6 9386.2.a.j.1.1 1
260.199 odd 6 5200.2.a.x.1.1 1
312.173 odd 6 7488.2.a.g.1.1 1
312.251 even 6 7488.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.a.a.1.1 ✓ 1 13.4 even 6
208.2.a.a.1.1 1 52.43 odd 6
234.2.a.e.1.1 1 39.17 odd 6
338.2.a.f.1.1 1 13.9 even 3
338.2.b.c.337.1 2 13.6 odd 12
338.2.b.c.337.2 2 13.7 odd 12
338.2.c.a.191.1 2 1.1 even 1 trivial
338.2.c.a.315.1 2 13.3 even 3 inner
338.2.c.d.191.1 2 13.12 even 2
338.2.c.d.315.1 2 13.10 even 6
338.2.e.a.23.1 4 13.2 odd 12
338.2.e.a.23.2 4 13.11 odd 12
338.2.e.a.147.1 4 13.8 odd 4
338.2.e.a.147.2 4 13.5 odd 4
650.2.a.j.1.1 1 65.4 even 6
650.2.b.d.599.1 2 65.17 odd 12
650.2.b.d.599.2 2 65.43 odd 12
832.2.a.d.1.1 1 104.69 even 6
832.2.a.i.1.1 1 104.43 odd 6
1274.2.a.d.1.1 1 91.69 odd 6
1274.2.f.p.79.1 2 91.4 even 6
1274.2.f.p.1145.1 2 91.30 even 6
1274.2.f.r.79.1 2 91.17 odd 6
1274.2.f.r.1145.1 2 91.82 odd 6
1872.2.a.q.1.1 1 156.95 even 6
2106.2.e.b.703.1 2 117.95 odd 6
2106.2.e.b.1405.1 2 117.56 odd 6
2106.2.e.ba.703.1 2 117.4 even 6
2106.2.e.ba.1405.1 2 117.43 even 6
2704.2.a.f.1.1 1 52.35 odd 6
2704.2.f.d.337.1 2 52.19 even 12
2704.2.f.d.337.2 2 52.7 even 12
3042.2.a.a.1.1 1 39.35 odd 6
3042.2.b.a.1351.1 2 39.20 even 12
3042.2.b.a.1351.2 2 39.32 even 12
3146.2.a.n.1.1 1 143.43 odd 6
3328.2.b.j.1665.1 2 208.147 odd 12
3328.2.b.j.1665.2 2 208.43 odd 12
3328.2.b.m.1665.1 2 208.69 even 12
3328.2.b.m.1665.2 2 208.173 even 12
5200.2.a.x.1.1 1 260.199 odd 6
5850.2.a.p.1.1 1 195.134 odd 6
5850.2.e.a.5149.1 2 195.173 even 12
5850.2.e.a.5149.2 2 195.17 even 12
7488.2.a.g.1.1 1 312.173 odd 6
7488.2.a.h.1.1 1 312.251 even 6
7514.2.a.c.1.1 1 221.186 even 6
8450.2.a.c.1.1 1 65.9 even 6
9386.2.a.j.1.1 1 247.56 odd 6