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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [338,3,Mod(19,338)] 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("338.19"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 338.f (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,12,8,0,16,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.20983293538\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 19.5
Character \(\chi\) \(=\) 338.19
Dual form 338.3.f.l.89.5

$q$-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+(-0.366025 - 1.36603i) q^{2} +(1.61556 - 2.79824i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(0.725610 - 0.725610i) q^{5} +(-4.41380 - 1.18267i) q^{6} +(2.54949 - 9.51482i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-0.720083 - 1.24722i) q^{9} +(-1.25679 - 0.725610i) q^{10} +(20.8198 - 5.57864i) q^{11} +6.46225i q^{12} -13.9307 q^{14} +(-0.858160 - 3.20270i) q^{15} +(2.00000 - 3.46410i) q^{16} +(-2.12172 + 1.22497i) q^{17} +(-1.44017 + 1.44017i) q^{18} +(-22.3552 - 5.99006i) q^{19} +(-0.531183 + 1.98240i) q^{20} +(-22.5059 - 22.5059i) q^{21} +(-15.2411 - 26.3984i) q^{22} +(-21.0534 - 12.1552i) q^{23} +(8.82760 - 2.36535i) q^{24} +23.9470i q^{25} +24.4268 q^{27} +(5.09898 + 19.0296i) q^{28} +(-14.4165 + 24.9701i) q^{29} +(-4.06086 + 2.34454i) q^{30} +(30.8436 - 30.8436i) q^{31} +(-5.46410 - 1.46410i) q^{32} +(18.0253 - 67.2713i) q^{33} +(2.44995 + 2.44995i) q^{34} +(-5.05411 - 8.75398i) q^{35} +(2.49444 + 1.44017i) q^{36} +(-46.7991 + 12.5398i) q^{37} +32.7303i q^{38} +2.90244 q^{40} +(6.72009 + 25.0797i) q^{41} +(-22.5059 + 38.9813i) q^{42} +(-5.79601 + 3.34633i) q^{43} +(-30.4823 + 30.4823i) q^{44} +(-1.42749 - 0.382496i) q^{45} +(-8.89821 + 33.2086i) q^{46} +(-9.85769 - 9.85769i) q^{47} +(-6.46225 - 11.1929i) q^{48} +(-41.5966 - 24.0158i) q^{49} +(32.7122 - 8.76520i) q^{50} +7.91608i q^{51} +36.3764 q^{53} +(-8.94082 - 33.3676i) q^{54} +(11.0591 - 19.1549i) q^{55} +(24.1286 - 13.9307i) q^{56} +(-52.8778 + 52.8778i) q^{57} +(39.3866 + 10.5536i) q^{58} +(-0.0722962 + 0.269813i) q^{59} +(4.68907 + 4.68907i) q^{60} +(-6.76799 - 11.7225i) q^{61} +(-53.4228 - 30.8436i) q^{62} +(-13.7029 + 3.67168i) q^{63} +8.00000i q^{64} -98.4919 q^{66} +(11.4887 + 42.8764i) q^{67} +(2.44995 - 4.24343i) q^{68} +(-68.0261 + 39.2749i) q^{69} +(-10.1082 + 10.1082i) q^{70} +(-48.2513 - 12.9289i) q^{71} +(1.05427 - 3.93460i) q^{72} +(62.1910 + 62.1910i) q^{73} +(34.2593 + 59.3389i) q^{74} +(67.0093 + 38.6878i) q^{75} +(44.7104 - 11.9801i) q^{76} -212.319i q^{77} +30.7872 q^{79} +(-1.06237 - 3.96481i) q^{80} +(45.9437 - 79.5768i) q^{81} +(31.7998 - 18.3596i) q^{82} +(84.7223 - 84.7223i) q^{83} +(61.4871 + 16.4754i) q^{84} +(-0.650685 + 2.42839i) q^{85} +(6.69265 + 6.69265i) q^{86} +(46.5815 + 80.6816i) q^{87} +(52.7968 + 30.4823i) q^{88} +(-67.9871 + 18.2171i) q^{89} +2.09000i q^{90} +48.6207 q^{92} +(-36.4780 - 136.138i) q^{93} +(-9.85769 + 17.0740i) q^{94} +(-20.5676 + 11.8747i) q^{95} +(-12.9245 + 12.9245i) q^{96} +(69.0727 + 18.5080i) q^{97} +(-17.5808 + 65.6125i) q^{98} +(-21.9497 - 21.9497i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 12 q^{2} + 8 q^{3} + 16 q^{5} - 8 q^{6} - 24 q^{7} + 48 q^{8} - 28 q^{9} - 16 q^{11} - 96 q^{14} - 116 q^{15} + 48 q^{16} - 56 q^{18} - 92 q^{19} + 16 q^{20} - 256 q^{21} + 32 q^{22} + 16 q^{24}+ \cdots + 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(e\left(\frac{5}{12}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.366025 − 1.36603i −0.183013 − 0.683013i
\(3\) 1.61556 − 2.79824i 0.538521 − 0.932745i −0.460463 − 0.887679i \(-0.652317\pi\)
0.998984 − 0.0450665i \(-0.0143500\pi\)
\(4\) −1.73205 + 1.00000i −0.433013 + 0.250000i
\(5\) 0.725610 − 0.725610i 0.145122 − 0.145122i −0.630813 − 0.775935i \(-0.717278\pi\)
0.775935 + 0.630813i \(0.217278\pi\)
\(6\) −4.41380 − 1.18267i −0.735633 − 0.197112i
\(7\) 2.54949 − 9.51482i 0.364213 − 1.35926i −0.504273 − 0.863545i \(-0.668239\pi\)
0.868485 − 0.495715i \(-0.165094\pi\)
\(8\) 2.00000 + 2.00000i 0.250000 + 0.250000i
\(9\) −0.720083 − 1.24722i −0.0800092 − 0.138580i
\(10\) −1.25679 − 0.725610i −0.125679 − 0.0725610i
\(11\) 20.8198 − 5.57864i 1.89271 − 0.507149i 0.894514 − 0.447040i \(-0.147522\pi\)
0.998192 − 0.0601093i \(-0.0191449\pi\)
\(12\) 6.46225i 0.538521i
\(13\) 0 0
\(14\) −13.9307 −0.995047
\(15\) −0.858160 − 3.20270i −0.0572106 − 0.213513i
\(16\) 2.00000 − 3.46410i 0.125000 − 0.216506i
\(17\) −2.12172 + 1.22497i −0.124807 + 0.0720573i −0.561104 − 0.827746i \(-0.689623\pi\)
0.436297 + 0.899803i \(0.356290\pi\)
\(18\) −1.44017 + 1.44017i −0.0800092 + 0.0800092i
\(19\) −22.3552 − 5.99006i −1.17659 − 0.315266i −0.383017 − 0.923741i \(-0.625115\pi\)
−0.793573 + 0.608475i \(0.791782\pi\)
\(20\) −0.531183 + 1.98240i −0.0265592 + 0.0991202i
\(21\) −22.5059 − 22.5059i −1.07171 − 1.07171i
\(22\) −15.2411 − 26.3984i −0.692778 − 1.19993i
\(23\) −21.0534 − 12.1552i −0.915365 − 0.528486i −0.0332115 − 0.999448i \(-0.510573\pi\)
−0.882153 + 0.470962i \(0.843907\pi\)
\(24\) 8.82760 − 2.36535i 0.367817 − 0.0985561i
\(25\) 23.9470i 0.957879i
\(26\) 0 0
\(27\) 24.4268 0.904695
\(28\) 5.09898 + 19.0296i 0.182106 + 0.679630i
\(29\) −14.4165 + 24.9701i −0.497121 + 0.861039i −0.999994 − 0.00332108i \(-0.998943\pi\)
0.502873 + 0.864360i \(0.332276\pi\)
\(30\) −4.06086 + 2.34454i −0.135362 + 0.0781512i
\(31\) 30.8436 − 30.8436i 0.994956 − 0.994956i −0.00503120 − 0.999987i \(-0.501601\pi\)
0.999987 + 0.00503120i \(0.00160149\pi\)
\(32\) −5.46410 − 1.46410i −0.170753 − 0.0457532i
\(33\) 18.0253 − 67.2713i 0.546221 − 2.03852i
\(34\) 2.44995 + 2.44995i 0.0720573 + 0.0720573i
\(35\) −5.05411 − 8.75398i −0.144403 − 0.250114i
\(36\) 2.49444 + 1.44017i 0.0692900 + 0.0400046i
\(37\) −46.7991 + 12.5398i −1.26484 + 0.338913i −0.828052 − 0.560651i \(-0.810551\pi\)
−0.436788 + 0.899564i \(0.643884\pi\)
\(38\) 32.7303i 0.861324i
\(39\) 0 0
\(40\) 2.90244 0.0725610
\(41\) 6.72009 + 25.0797i 0.163905 + 0.611701i 0.998177 + 0.0603487i \(0.0192213\pi\)
−0.834273 + 0.551352i \(0.814112\pi\)
\(42\) −22.5059 + 38.9813i −0.535854 + 0.928126i
\(43\) −5.79601 + 3.34633i −0.134791 + 0.0778215i −0.565879 − 0.824488i \(-0.691463\pi\)
0.431088 + 0.902310i \(0.358130\pi\)
\(44\) −30.4823 + 30.4823i −0.692778 + 0.692778i
\(45\) −1.42749 − 0.382496i −0.0317221 − 0.00849991i
\(46\) −8.89821 + 33.2086i −0.193439 + 0.721926i
\(47\) −9.85769 − 9.85769i −0.209738 − 0.209738i 0.594418 − 0.804156i \(-0.297382\pi\)
−0.804156 + 0.594418i \(0.797382\pi\)
\(48\) −6.46225 − 11.1929i −0.134630 − 0.233186i
\(49\) −41.5966 − 24.0158i −0.848911 − 0.490119i
\(50\) 32.7122 − 8.76520i 0.654244 − 0.175304i
\(51\) 7.91608i 0.155217i
\(52\) 0 0
\(53\) 36.3764 0.686348 0.343174 − 0.939272i \(-0.388498\pi\)
0.343174 + 0.939272i \(0.388498\pi\)
\(54\) −8.94082 − 33.3676i −0.165571 − 0.617918i
\(55\) 11.0591 − 19.1549i 0.201075 − 0.348272i
\(56\) 24.1286 − 13.9307i 0.430868 − 0.248762i
\(57\) −52.8778 + 52.8778i −0.927681 + 0.927681i
\(58\) 39.3866 + 10.5536i 0.679080 + 0.181959i
\(59\) −0.0722962 + 0.269813i −0.00122536 + 0.00457310i −0.966536 − 0.256532i \(-0.917420\pi\)
0.965310 + 0.261105i \(0.0840868\pi\)
\(60\) 4.68907 + 4.68907i 0.0781512 + 0.0781512i
\(61\) −6.76799 − 11.7225i −0.110951 − 0.192172i 0.805203 − 0.592999i \(-0.202056\pi\)
−0.916154 + 0.400827i \(0.868723\pi\)
\(62\) −53.4228 − 30.8436i −0.861657 − 0.497478i
\(63\) −13.7029 + 3.67168i −0.217507 + 0.0582807i
\(64\) 8.00000i 0.125000i
\(65\) 0 0
\(66\) −98.4919 −1.49230
\(67\) 11.4887 + 42.8764i 0.171473 + 0.639946i 0.997126 + 0.0757673i \(0.0241406\pi\)
−0.825652 + 0.564179i \(0.809193\pi\)
\(68\) 2.44995 − 4.24343i 0.0360286 − 0.0624034i
\(69\) −68.0261 + 39.2749i −0.985886 + 0.569202i
\(70\) −10.1082 + 10.1082i −0.144403 + 0.144403i
\(71\) −48.2513 − 12.9289i −0.679596 − 0.182097i −0.0975222 − 0.995233i \(-0.531092\pi\)
−0.582073 + 0.813136i \(0.697758\pi\)
\(72\) 1.05427 − 3.93460i 0.0146427 − 0.0546473i
\(73\) 62.1910 + 62.1910i 0.851931 + 0.851931i 0.990371 − 0.138439i \(-0.0442086\pi\)
−0.138439 + 0.990371i \(0.544209\pi\)
\(74\) 34.2593 + 59.3389i 0.462964 + 0.801876i
\(75\) 67.0093 + 38.6878i 0.893457 + 0.515838i
\(76\) 44.7104 − 11.9801i 0.588295 − 0.157633i
\(77\) − 212.319i − 2.75739i
\(78\) 0 0
\(79\) 30.7872 0.389712 0.194856 − 0.980832i \(-0.437576\pi\)
0.194856 + 0.980832i \(0.437576\pi\)
\(80\) −1.06237 − 3.96481i −0.0132796 − 0.0495601i
\(81\) 45.9437 − 79.5768i 0.567206 − 0.982430i
\(82\) 31.7998 − 18.3596i 0.387803 − 0.223898i
\(83\) 84.7223 − 84.7223i 1.02075 − 1.02075i 0.0209708 − 0.999780i \(-0.493324\pi\)
0.999780 − 0.0209708i \(-0.00667571\pi\)
\(84\) 61.4871 + 16.4754i 0.731990 + 0.196136i
\(85\) −0.650685 + 2.42839i −0.00765512 + 0.0285693i
\(86\) 6.69265 + 6.69265i 0.0778215 + 0.0778215i
\(87\) 46.5815 + 80.6816i 0.535420 + 0.927375i
\(88\) 52.7968 + 30.4823i 0.599964 + 0.346389i
\(89\) −67.9871 + 18.2171i −0.763900 + 0.204686i −0.619675 − 0.784859i \(-0.712736\pi\)
−0.144225 + 0.989545i \(0.546069\pi\)
\(90\) 2.09000i 0.0232222i
\(91\) 0 0
\(92\) 48.6207 0.528486
\(93\) −36.4780 − 136.138i −0.392236 − 1.46385i
\(94\) −9.85769 + 17.0740i −0.104869 + 0.181639i
\(95\) −20.5676 + 11.8747i −0.216501 + 0.124997i
\(96\) −12.9245 + 12.9245i −0.134630 + 0.134630i
\(97\) 69.0727 + 18.5080i 0.712089 + 0.190804i 0.596639 − 0.802510i \(-0.296502\pi\)
0.115450 + 0.993313i \(0.463169\pi\)
\(98\) −17.5808 + 65.6125i −0.179396 + 0.669515i
\(99\) −21.9497 − 21.9497i −0.221715 − 0.221715i
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 338.3.f.l.19.5 24
13.2 odd 12 338.3.f.k.89.5 24
13.3 even 3 inner 338.3.f.l.249.5 24
13.4 even 6 338.3.d.i.239.2 yes 12
13.5 odd 4 338.3.f.k.319.5 24
13.6 odd 12 338.3.d.i.99.2 yes 12
13.7 odd 12 338.3.d.h.99.2 ✓ 12
13.8 odd 4 inner 338.3.f.l.319.5 24
13.9 even 3 338.3.d.h.239.2 yes 12
13.10 even 6 338.3.f.k.249.5 24
13.11 odd 12 inner 338.3.f.l.89.5 24
13.12 even 2 338.3.f.k.19.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
338.3.d.h.99.2 ✓ 12 13.7 odd 12
338.3.d.h.239.2 yes 12 13.9 even 3
338.3.d.i.99.2 yes 12 13.6 odd 12
338.3.d.i.239.2 yes 12 13.4 even 6
338.3.f.k.19.5 24 13.12 even 2
338.3.f.k.89.5 24 13.2 odd 12
338.3.f.k.249.5 24 13.10 even 6
338.3.f.k.319.5 24 13.5 odd 4
338.3.f.l.19.5 24 1.1 even 1 trivial
338.3.f.l.89.5 24 13.11 odd 12 inner
338.3.f.l.249.5 24 13.3 even 3 inner
338.3.f.l.319.5 24 13.8 odd 4 inner