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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 319.5
Character \(\chi\) \(=\) 338.319
Dual form 338.3.f.l.249.5

$q$-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+(1.36603 - 0.366025i) q^{2} +(1.61556 - 2.79824i) q^{3} +(1.73205 - 1.00000i) q^{4} +(0.725610 + 0.725610i) q^{5} +(1.18267 - 4.41380i) q^{6} +(-9.51482 - 2.54949i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-0.720083 - 1.24722i) q^{9} +(1.25679 + 0.725610i) q^{10} +(-5.57864 - 20.8198i) q^{11} -6.46225i q^{12} -13.9307 q^{14} +(3.20270 - 0.858160i) q^{15} +(2.00000 - 3.46410i) q^{16} +(2.12172 - 1.22497i) q^{17} +(-1.44017 - 1.44017i) q^{18} +(5.99006 - 22.3552i) q^{19} +(1.98240 + 0.531183i) q^{20} +(-22.5059 + 22.5059i) q^{21} +(-15.2411 - 26.3984i) q^{22} +(21.0534 + 12.1552i) q^{23} +(-2.36535 - 8.82760i) q^{24} -23.9470i q^{25} +24.4268 q^{27} +(-19.0296 + 5.09898i) q^{28} +(-14.4165 + 24.9701i) q^{29} +(4.06086 - 2.34454i) q^{30} +(30.8436 + 30.8436i) q^{31} +(1.46410 - 5.46410i) q^{32} +(-67.2713 - 18.0253i) q^{33} +(2.44995 - 2.44995i) q^{34} +(-5.05411 - 8.75398i) q^{35} +(-2.49444 - 1.44017i) q^{36} +(12.5398 + 46.7991i) q^{37} -32.7303i q^{38} +2.90244 q^{40} +(-25.0797 + 6.72009i) q^{41} +(-22.5059 + 38.9813i) q^{42} +(5.79601 - 3.34633i) q^{43} +(-30.4823 - 30.4823i) q^{44} +(0.382496 - 1.42749i) q^{45} +(33.2086 + 8.89821i) q^{46} +(-9.85769 + 9.85769i) q^{47} +(-6.46225 - 11.1929i) q^{48} +(41.5966 + 24.0158i) q^{49} +(-8.76520 - 32.7122i) q^{50} -7.91608i q^{51} +36.3764 q^{53} +(33.3676 - 8.94082i) q^{54} +(11.0591 - 19.1549i) q^{55} +(-24.1286 + 13.9307i) q^{56} +(-52.8778 - 52.8778i) q^{57} +(-10.5536 + 39.3866i) q^{58} +(0.269813 + 0.0722962i) q^{59} +(4.68907 - 4.68907i) q^{60} +(-6.76799 - 11.7225i) q^{61} +(53.4228 + 30.8436i) q^{62} +(3.67168 + 13.7029i) q^{63} -8.00000i q^{64} -98.4919 q^{66} +(-42.8764 + 11.4887i) q^{67} +(2.44995 - 4.24343i) q^{68} +(68.0261 - 39.2749i) q^{69} +(-10.1082 - 10.1082i) q^{70} +(12.9289 - 48.2513i) q^{71} +(-3.93460 - 1.05427i) q^{72} +(62.1910 - 62.1910i) q^{73} +(34.2593 + 59.3389i) q^{74} +(-67.0093 - 38.6878i) q^{75} +(-11.9801 - 44.7104i) q^{76} +212.319i q^{77} +30.7872 q^{79} +(3.96481 - 1.06237i) q^{80} +(45.9437 - 79.5768i) q^{81} +(-31.7998 + 18.3596i) q^{82} +(84.7223 + 84.7223i) q^{83} +(-16.4754 + 61.4871i) q^{84} +(2.42839 + 0.650685i) q^{85} +(6.69265 - 6.69265i) q^{86} +(46.5815 + 80.6816i) q^{87} +(-52.7968 - 30.4823i) q^{88} +(18.2171 + 67.9871i) q^{89} -2.09000i q^{90} +48.6207 q^{92} +(136.138 - 36.4780i) q^{93} +(-9.85769 + 17.0740i) q^{94} +(20.5676 - 11.8747i) q^{95} +(-12.9245 - 12.9245i) q^{96} +(-18.5080 + 69.0727i) q^{97} +(65.6125 + 17.5808i) 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 \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(e\left(\frac{11}{12}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36603 − 0.366025i 0.683013 − 0.183013i
\(3\) 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 0.775935 − 0.630813i \(-0.217278\pi\)
−0.630813 + 0.775935i \(0.717278\pi\)
\(6\) 1.18267 − 4.41380i 0.197112 − 0.735633i
\(7\) −9.51482 − 2.54949i −1.35926 − 0.364213i −0.495715 − 0.868485i \(-0.665094\pi\)
−0.863545 + 0.504273i \(0.831761\pi\)
\(8\) 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
\(11\) −5.57864 − 20.8198i −0.507149 − 1.89271i −0.447040 − 0.894514i \(-0.647522\pi\)
−0.0601093 − 0.998192i \(-0.519145\pi\)
\(12\) − 6.46225i − 0.538521i
\(13\) 0 0
\(14\) −13.9307 −0.995047
\(15\) 3.20270 − 0.858160i 0.213513 − 0.0572106i
\(16\) 2.00000 − 3.46410i 0.125000 − 0.216506i
\(17\) 2.12172 − 1.22497i 0.124807 − 0.0720573i −0.436297 − 0.899803i \(-0.643710\pi\)
0.561104 + 0.827746i \(0.310377\pi\)
\(18\) −1.44017 − 1.44017i −0.0800092 − 0.0800092i
\(19\) 5.99006 − 22.3552i 0.315266 − 1.17659i −0.608475 − 0.793573i \(-0.708218\pi\)
0.923741 − 0.383017i \(-0.125115\pi\)
\(20\) 1.98240 + 0.531183i 0.0991202 + 0.0265592i
\(21\) −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 0.882153 − 0.470962i \(-0.156093\pi\)
0.0332115 + 0.999448i \(0.489427\pi\)
\(24\) −2.36535 − 8.82760i −0.0985561 − 0.367817i
\(25\) − 23.9470i − 0.957879i
\(26\) 0 0
\(27\) 24.4268 0.904695
\(28\) −19.0296 + 5.09898i −0.679630 + 0.182106i
\(29\) −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 0.999987 − 0.00503120i \(-0.00160149\pi\)
−0.00503120 + 0.999987i \(0.501601\pi\)
\(32\) 1.46410 − 5.46410i 0.0457532 − 0.170753i
\(33\) −67.2713 − 18.0253i −2.03852 − 0.546221i
\(34\) 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
\(37\) 12.5398 + 46.7991i 0.338913 + 1.26484i 0.899564 + 0.436788i \(0.143884\pi\)
−0.560651 + 0.828052i \(0.689449\pi\)
\(38\) − 32.7303i − 0.861324i
\(39\) 0 0
\(40\) 2.90244 0.0725610
\(41\) −25.0797 + 6.72009i −0.611701 + 0.163905i −0.551352 − 0.834273i \(-0.685888\pi\)
−0.0603487 + 0.998177i \(0.519221\pi\)
\(42\) −22.5059 + 38.9813i −0.535854 + 0.928126i
\(43\) 5.79601 − 3.34633i 0.134791 − 0.0778215i −0.431088 − 0.902310i \(-0.641870\pi\)
0.565879 + 0.824488i \(0.308537\pi\)
\(44\) −30.4823 − 30.4823i −0.692778 − 0.692778i
\(45\) 0.382496 − 1.42749i 0.00849991 − 0.0317221i
\(46\) 33.2086 + 8.89821i 0.721926 + 0.193439i
\(47\) −9.85769 + 9.85769i −0.209738 + 0.209738i −0.804156 − 0.594418i \(-0.797382\pi\)
0.594418 + 0.804156i \(0.297382\pi\)
\(48\) −6.46225 − 11.1929i −0.134630 − 0.233186i
\(49\) 41.5966 + 24.0158i 0.848911 + 0.490119i
\(50\) −8.76520 − 32.7122i −0.175304 − 0.654244i
\(51\) − 7.91608i − 0.155217i
\(52\) 0 0
\(53\) 36.3764 0.686348 0.343174 − 0.939272i \(-0.388498\pi\)
0.343174 + 0.939272i \(0.388498\pi\)
\(54\) 33.3676 − 8.94082i 0.617918 − 0.165571i
\(55\) 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
\(58\) −10.5536 + 39.3866i −0.181959 + 0.679080i
\(59\) 0.269813 + 0.0722962i 0.00457310 + 0.00122536i 0.261105 − 0.965310i \(-0.415913\pi\)
−0.256532 + 0.966536i \(0.582580\pi\)
\(60\) 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
\(63\) 3.67168 + 13.7029i 0.0582807 + 0.217507i
\(64\) − 8.00000i − 0.125000i
\(65\) 0 0
\(66\) −98.4919 −1.49230
\(67\) −42.8764 + 11.4887i −0.639946 + 0.171473i −0.564179 − 0.825652i \(-0.690807\pi\)
−0.0757673 + 0.997126i \(0.524141\pi\)
\(68\) 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
\(71\) 12.9289 − 48.2513i 0.182097 − 0.679596i −0.813136 − 0.582073i \(-0.802242\pi\)
0.995233 − 0.0975222i \(-0.0310917\pi\)
\(72\) −3.93460 − 1.05427i −0.0546473 − 0.0146427i
\(73\) 62.1910 − 62.1910i 0.851931 − 0.851931i −0.138439 − 0.990371i \(-0.544209\pi\)
0.990371 + 0.138439i \(0.0442086\pi\)
\(74\) 34.2593 + 59.3389i 0.462964 + 0.801876i
\(75\) −67.0093 − 38.6878i −0.893457 − 0.515838i
\(76\) −11.9801 − 44.7104i −0.157633 − 0.588295i
\(77\) 212.319i 2.75739i
\(78\) 0 0
\(79\) 30.7872 0.389712 0.194856 − 0.980832i \(-0.437576\pi\)
0.194856 + 0.980832i \(0.437576\pi\)
\(80\) 3.96481 − 1.06237i 0.0495601 − 0.0132796i
\(81\) 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 0.999780 + 0.0209708i \(0.00667571\pi\)
0.0209708 + 0.999780i \(0.493324\pi\)
\(84\) −16.4754 + 61.4871i −0.196136 + 0.731990i
\(85\) 2.42839 + 0.650685i 0.0285693 + 0.00765512i
\(86\) 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
\(89\) 18.2171 + 67.9871i 0.204686 + 0.763900i 0.989545 + 0.144225i \(0.0460689\pi\)
−0.784859 + 0.619675i \(0.787264\pi\)
\(90\) − 2.09000i − 0.0232222i
\(91\) 0 0
\(92\) 48.6207 0.528486
\(93\) 136.138 − 36.4780i 1.46385 − 0.392236i
\(94\) −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
\(97\) −18.5080 + 69.0727i −0.190804 + 0.712089i 0.802510 + 0.596639i \(0.203498\pi\)
−0.993313 + 0.115450i \(0.963169\pi\)
\(98\) 65.6125 + 17.5808i 0.669515 + 0.179396i
\(99\) −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.319.5 24
13.2 odd 12 inner 338.3.f.l.249.5 24
13.3 even 3 inner 338.3.f.l.89.5 24
13.4 even 6 338.3.d.i.99.2 yes 12
13.5 odd 4 inner 338.3.f.l.19.5 24
13.6 odd 12 338.3.d.h.239.2 yes 12
13.7 odd 12 338.3.d.i.239.2 yes 12
13.8 odd 4 338.3.f.k.19.5 24
13.9 even 3 338.3.d.h.99.2 ✓ 12
13.10 even 6 338.3.f.k.89.5 24
13.11 odd 12 338.3.f.k.249.5 24
13.12 even 2 338.3.f.k.319.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
338.3.d.h.99.2 ✓ 12 13.9 even 3
338.3.d.h.239.2 yes 12 13.6 odd 12
338.3.d.i.99.2 yes 12 13.4 even 6
338.3.d.i.239.2 yes 12 13.7 odd 12
338.3.f.k.19.5 24 13.8 odd 4
338.3.f.k.89.5 24 13.10 even 6
338.3.f.k.249.5 24 13.11 odd 12
338.3.f.k.319.5 24 13.12 even 2
338.3.f.l.19.5 24 13.5 odd 4 inner
338.3.f.l.89.5 24 13.3 even 3 inner
338.3.f.l.249.5 24 13.2 odd 12 inner
338.3.f.l.319.5 24 1.1 even 1 trivial