Show commands: Magma / Pari/GP / SageMath

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 = [8,4,0,0,-6,6] 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.612074651904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 89.2
Root \(4.71318 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 338.89
Dual form 338.3.f.j.19.2

$q$-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} +(1.92358 + 3.33174i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-3.77418 - 3.77418i) q^{5} +(-5.25532 + 1.40816i) q^{6} +(-2.65563 - 9.91095i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-2.90031 + 5.02349i) q^{9} +(6.53708 - 3.77418i) q^{10} +(10.1197 + 2.71157i) q^{11} -7.69432i q^{12} +14.5106 q^{14} +(5.31465 - 19.8345i) q^{15} +(2.00000 + 3.46410i) q^{16} +(-4.23323 - 2.44406i) q^{17} +(-5.80063 - 5.80063i) q^{18} +(25.5153 - 6.83679i) q^{19} +(2.76289 + 10.3113i) q^{20} +(27.9124 - 27.9124i) q^{21} +(-7.40816 + 12.8313i) q^{22} +(17.2850 - 9.97952i) q^{23} +(10.5106 + 2.81632i) q^{24} +3.48892i q^{25} +12.3085 q^{27} +(-5.31126 + 19.8219i) q^{28} +(-7.15218 - 12.3879i) q^{29} +(25.1492 + 14.5199i) q^{30} +(-19.0056 - 19.0056i) q^{31} +(-5.46410 + 1.46410i) q^{32} +(10.4319 + 38.9322i) q^{33} +(4.88811 - 4.88811i) q^{34} +(-27.3829 + 47.4286i) q^{35} +(10.0470 - 5.80063i) q^{36} +(-58.6123 - 15.7051i) q^{37} +37.3569i q^{38} -15.0967 q^{40} +(-1.29609 + 4.83709i) q^{41} +(27.9124 + 48.3456i) q^{42} +(10.3688 + 5.98641i) q^{43} +(-14.8163 - 14.8163i) q^{44} +(29.9059 - 8.01326i) q^{45} +(7.30552 + 27.2646i) q^{46} +(7.59168 - 7.59168i) q^{47} +(-7.69432 + 13.3269i) q^{48} +(-48.7392 + 28.1396i) q^{49} +(-4.76595 - 1.27703i) q^{50} -18.8053i q^{51} +77.0450 q^{53} +(-4.50522 + 16.8137i) q^{54} +(-27.9597 - 48.4277i) q^{55} +(-25.1332 - 14.5106i) q^{56} +(71.8590 + 71.8590i) q^{57} +(19.5401 - 5.23576i) q^{58} +(-16.2815 - 60.7634i) q^{59} +(-29.0398 + 29.0398i) q^{60} +(28.1382 - 48.7368i) q^{61} +(32.9186 - 19.0056i) q^{62} +(57.4897 + 15.4043i) q^{63} -8.00000i q^{64} -57.0007 q^{66} +(1.58199 - 5.90406i) q^{67} +(4.88811 + 8.46646i) q^{68} +(66.4983 + 38.3928i) q^{69} +(-54.7658 - 54.7658i) q^{70} +(-55.2272 + 14.7981i) q^{71} +(4.24635 + 15.8476i) q^{72} +(-12.7990 + 12.7990i) q^{73} +(42.9072 - 74.3174i) q^{74} +(-11.6242 + 6.71121i) q^{75} +(-51.0305 - 13.6736i) q^{76} -107.497i q^{77} +7.98532 q^{79} +(5.52579 - 20.6225i) q^{80} +(49.7792 + 86.2201i) q^{81} +(-6.13318 - 3.54099i) q^{82} +(35.8343 + 35.8343i) q^{83} +(-76.2580 + 20.4333i) q^{84} +(6.75267 + 25.2013i) q^{85} +(-11.9728 + 11.9728i) q^{86} +(27.5156 - 47.6584i) q^{87} +(25.6626 - 14.8163i) q^{88} +(78.2551 + 20.9684i) q^{89} +43.7853i q^{90} -39.9181 q^{92} +(26.7628 - 99.8803i) q^{93} +(7.59168 + 13.1492i) q^{94} +(-122.103 - 70.4959i) q^{95} +(-15.3886 - 15.3886i) q^{96} +(-52.9919 + 14.1991i) q^{97} +(-20.5996 - 76.8788i) q^{98} +(-42.9720 + 42.9720i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 6 q^{5} + 6 q^{6} + 8 q^{7} + 16 q^{8} - 42 q^{9} + 18 q^{10} + 24 q^{11} + 20 q^{14} + 126 q^{15} + 16 q^{16} + 42 q^{17} - 84 q^{18} + 68 q^{19} + 12 q^{20} + 102 q^{21} - 42 q^{22} + 36 q^{23}+ \cdots - 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(e\left(\frac{7}{12}\right)\)

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
\(3\) 1.92358 + 3.33174i 0.641193 + 1.11058i 0.985167 + 0.171600i \(0.0548936\pi\)
−0.343974 + 0.938979i \(0.611773\pi\)
\(4\) −1.73205 − 1.00000i −0.433013 − 0.250000i
\(5\) −3.77418 − 3.77418i −0.754837 − 0.754837i 0.220541 − 0.975378i \(-0.429218\pi\)
−0.975378 + 0.220541i \(0.929218\pi\)
\(6\) −5.25532 + 1.40816i −0.875886 + 0.234693i
\(7\) −2.65563 − 9.91095i −0.379376 − 1.41585i −0.846845 − 0.531840i \(-0.821501\pi\)
0.467469 − 0.884009i \(-0.345166\pi\)
\(8\) 2.00000 − 2.00000i 0.250000 − 0.250000i
\(9\) −2.90031 + 5.02349i −0.322257 + 0.558166i
\(10\) 6.53708 − 3.77418i 0.653708 − 0.377418i
\(11\) 10.1197 + 2.71157i 0.919976 + 0.246507i 0.687575 − 0.726113i \(-0.258675\pi\)
0.232401 + 0.972620i \(0.425342\pi\)
\(12\) − 7.69432i − 0.641193i
\(13\) 0 0
\(14\) 14.5106 1.03647
\(15\) 5.31465 − 19.8345i 0.354310 − 1.32230i
\(16\) 2.00000 + 3.46410i 0.125000 + 0.216506i
\(17\) −4.23323 − 2.44406i −0.249013 − 0.143768i 0.370299 − 0.928913i \(-0.379255\pi\)
−0.619312 + 0.785145i \(0.712589\pi\)
\(18\) −5.80063 − 5.80063i −0.322257 − 0.322257i
\(19\) 25.5153 − 6.83679i 1.34291 − 0.359831i 0.485396 − 0.874295i \(-0.338675\pi\)
0.857512 + 0.514463i \(0.172009\pi\)
\(20\) 2.76289 + 10.3113i 0.138145 + 0.515563i
\(21\) 27.9124 − 27.9124i 1.32916 − 1.32916i
\(22\) −7.40816 + 12.8313i −0.336734 + 0.583241i
\(23\) 17.2850 − 9.97952i 0.751524 − 0.433892i −0.0747206 − 0.997205i \(-0.523806\pi\)
0.826244 + 0.563312i \(0.190473\pi\)
\(24\) 10.5106 + 2.81632i 0.437943 + 0.117346i
\(25\) 3.48892i 0.139557i
\(26\) 0 0
\(27\) 12.3085 0.455870
\(28\) −5.31126 + 19.8219i −0.189688 + 0.707925i
\(29\) −7.15218 − 12.3879i −0.246627 − 0.427171i 0.715961 − 0.698141i \(-0.245989\pi\)
−0.962588 + 0.270970i \(0.912656\pi\)
\(30\) 25.1492 + 14.5199i 0.838306 + 0.483996i
\(31\) −19.0056 − 19.0056i −0.613083 − 0.613083i 0.330665 − 0.943748i \(-0.392727\pi\)
−0.943748 + 0.330665i \(0.892727\pi\)
\(32\) −5.46410 + 1.46410i −0.170753 + 0.0457532i
\(33\) 10.4319 + 38.9322i 0.316117 + 1.17976i
\(34\) 4.88811 − 4.88811i 0.143768 − 0.143768i
\(35\) −27.3829 + 47.4286i −0.782368 + 1.35510i
\(36\) 10.0470 − 5.80063i 0.279083 − 0.161129i
\(37\) −58.6123 − 15.7051i −1.58412 − 0.424463i −0.643919 − 0.765094i \(-0.722693\pi\)
−0.940197 + 0.340631i \(0.889359\pi\)
\(38\) 37.3569i 0.983077i
\(39\) 0 0
\(40\) −15.0967 −0.377418
\(41\) −1.29609 + 4.83709i −0.0316120 + 0.117978i −0.979929 − 0.199347i \(-0.936118\pi\)
0.948317 + 0.317325i \(0.102785\pi\)
\(42\) 27.9124 + 48.3456i 0.664580 + 1.15109i
\(43\) 10.3688 + 5.98641i 0.241134 + 0.139219i 0.615698 − 0.787982i \(-0.288874\pi\)
−0.374564 + 0.927201i \(0.622207\pi\)
\(44\) −14.8163 − 14.8163i −0.336734 − 0.336734i
\(45\) 29.9059 − 8.01326i 0.664575 − 0.178072i
\(46\) 7.30552 + 27.2646i 0.158816 + 0.592708i
\(47\) 7.59168 − 7.59168i 0.161525 − 0.161525i −0.621717 − 0.783242i \(-0.713565\pi\)
0.783242 + 0.621717i \(0.213565\pi\)
\(48\) −7.69432 + 13.3269i −0.160298 + 0.277645i
\(49\) −48.7392 + 28.1396i −0.994678 + 0.574278i
\(50\) −4.76595 − 1.27703i −0.0953190 − 0.0255406i
\(51\) − 18.8053i − 0.368732i
\(52\) 0 0
\(53\) 77.0450 1.45368 0.726840 − 0.686807i \(-0.240988\pi\)
0.726840 + 0.686807i \(0.240988\pi\)
\(54\) −4.50522 + 16.8137i −0.0834300 + 0.311365i
\(55\) −27.9597 − 48.4277i −0.508359 − 0.880504i
\(56\) −25.1332 − 14.5106i −0.448806 − 0.259118i
\(57\) 71.8590 + 71.8590i 1.26068 + 1.26068i
\(58\) 19.5401 − 5.23576i 0.336899 − 0.0902718i
\(59\) −16.2815 − 60.7634i −0.275957 − 1.02989i −0.955197 − 0.295971i \(-0.904357\pi\)
0.679240 − 0.733917i \(-0.262310\pi\)
\(60\) −29.0398 + 29.0398i −0.483996 + 0.483996i
\(61\) 28.1382 − 48.7368i 0.461282 − 0.798964i −0.537743 − 0.843109i \(-0.680723\pi\)
0.999025 + 0.0441448i \(0.0140563\pi\)
\(62\) 32.9186 − 19.0056i 0.530946 − 0.306542i
\(63\) 57.4897 + 15.4043i 0.912535 + 0.244513i
\(64\) − 8.00000i − 0.125000i
\(65\) 0 0
\(66\) −57.0007 −0.863647
\(67\) 1.58199 − 5.90406i 0.0236117 − 0.0881202i −0.953115 − 0.302610i \(-0.902142\pi\)
0.976726 + 0.214490i \(0.0688088\pi\)
\(68\) 4.88811 + 8.46646i 0.0718840 + 0.124507i
\(69\) 66.4983 + 38.3928i 0.963743 + 0.556417i
\(70\) −54.7658 − 54.7658i −0.782368 − 0.782368i
\(71\) −55.2272 + 14.7981i −0.777847 + 0.208424i −0.625836 − 0.779955i \(-0.715242\pi\)
−0.152012 + 0.988379i \(0.548575\pi\)
\(72\) 4.24635 + 15.8476i 0.0589771 + 0.220106i
\(73\) −12.7990 + 12.7990i −0.175329 + 0.175329i −0.789316 − 0.613987i \(-0.789565\pi\)
0.613987 + 0.789316i \(0.289565\pi\)
\(74\) 42.9072 − 74.3174i 0.579827 − 1.00429i
\(75\) −11.6242 + 6.71121i −0.154989 + 0.0894828i
\(76\) −51.0305 − 13.6736i −0.671454 − 0.179916i
\(77\) − 107.497i − 1.39607i
\(78\) 0 0
\(79\) 7.98532 0.101080 0.0505400 − 0.998722i \(-0.483906\pi\)
0.0505400 + 0.998722i \(0.483906\pi\)
\(80\) 5.52579 − 20.6225i 0.0690723 − 0.257782i
\(81\) 49.7792 + 86.2201i 0.614558 + 1.06445i
\(82\) −6.13318 − 3.54099i −0.0747949 − 0.0431829i
\(83\) 35.8343 + 35.8343i 0.431738 + 0.431738i 0.889219 − 0.457481i \(-0.151248\pi\)
−0.457481 + 0.889219i \(0.651248\pi\)
\(84\) −76.2580 + 20.4333i −0.907833 + 0.243253i
\(85\) 6.75267 + 25.2013i 0.0794431 + 0.296486i
\(86\) −11.9728 + 11.9728i −0.139219 + 0.139219i
\(87\) 27.5156 − 47.6584i 0.316271 − 0.547798i
\(88\) 25.6626 − 14.8163i 0.291621 − 0.168367i
\(89\) 78.2551 + 20.9684i 0.879271 + 0.235600i 0.670093 − 0.742277i \(-0.266254\pi\)
0.209178 + 0.977877i \(0.432921\pi\)
\(90\) 43.7853i 0.486503i
\(91\) 0 0
\(92\) −39.9181 −0.433892
\(93\) 26.7628 − 99.8803i 0.287773 − 1.07398i
\(94\) 7.59168 + 13.1492i 0.0807625 + 0.139885i
\(95\) −122.103 − 70.4959i −1.28529 − 0.742063i
\(96\) −15.3886 − 15.3886i −0.160298 − 0.160298i
\(97\) −52.9919 + 14.1991i −0.546309 + 0.146383i −0.521409 − 0.853307i \(-0.674593\pi\)
−0.0248998 + 0.999690i \(0.507927\pi\)
\(98\) −20.5996 − 76.8788i −0.210200 − 0.784478i
\(99\) −42.9720 + 42.9720i −0.434060 + 0.434060i
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.j.89.2 8
13.2 odd 12 338.3.d.f.239.2 8
13.3 even 3 338.3.d.f.99.2 8
13.4 even 6 26.3.f.b.7.2 ✓ 8
13.5 odd 4 338.3.f.i.249.2 8
13.6 odd 12 inner 338.3.f.j.19.2 8
13.7 odd 12 338.3.f.h.19.2 8
13.8 odd 4 26.3.f.b.15.2 yes 8
13.9 even 3 338.3.f.i.319.2 8
13.10 even 6 338.3.d.g.99.2 8
13.11 odd 12 338.3.d.g.239.2 8
13.12 even 2 338.3.f.h.89.2 8
39.8 even 4 234.3.bb.f.145.1 8
39.17 odd 6 234.3.bb.f.163.1 8
52.43 odd 6 208.3.bd.f.33.1 8
52.47 even 4 208.3.bd.f.145.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.3.f.b.7.2 ✓ 8 13.4 even 6
26.3.f.b.15.2 yes 8 13.8 odd 4
208.3.bd.f.33.1 8 52.43 odd 6
208.3.bd.f.145.1 8 52.47 even 4
234.3.bb.f.145.1 8 39.8 even 4
234.3.bb.f.163.1 8 39.17 odd 6
338.3.d.f.99.2 8 13.3 even 3
338.3.d.f.239.2 8 13.2 odd 12
338.3.d.g.99.2 8 13.10 even 6
338.3.d.g.239.2 8 13.11 odd 12
338.3.f.h.19.2 8 13.7 odd 12
338.3.f.h.89.2 8 13.12 even 2
338.3.f.i.249.2 8 13.5 odd 4
338.3.f.i.319.2 8 13.9 even 3
338.3.f.j.19.2 8 13.6 odd 12 inner
338.3.f.j.89.2 8 1.1 even 1 trivial