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Properties

Label 42.6.e.c.37.2
Level $42$
Weight $6$
Character 42.37
Analytic conductor $6.736$
Analytic rank $0$
Dimension $4$
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] = [42,6,Mod(25,42)] 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("42.25"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.73612043215\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{9601})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.2
Root \(-24.2462 + 41.9956i\) of defining polynomial
Character \(\chi\) \(=\) 42.37
Dual form 42.6.e.c.25.2

$q$-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+(-2.00000 + 3.46410i) q^{2} +(4.50000 + 7.79423i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(37.7462 - 65.3783i) q^{5} -36.0000 q^{6} +(99.4847 - 83.1252i) q^{7} +64.0000 q^{8} +(-40.5000 + 70.1481i) q^{9} +(150.985 + 261.513i) q^{10} +(74.7309 + 129.438i) q^{11} +(72.0000 - 124.708i) q^{12} +349.416 q^{13} +(88.9847 + 510.875i) q^{14} +679.431 q^{15} +(-128.000 + 221.703i) q^{16} +(574.923 + 995.797i) q^{17} +(-162.000 - 280.592i) q^{18} +(1397.60 - 2420.72i) q^{19} -1207.88 q^{20} +(1095.58 + 401.343i) q^{21} -597.847 q^{22} +(-906.985 + 1570.94i) q^{23} +(288.000 + 498.831i) q^{24} +(-1287.05 - 2229.23i) q^{25} +(-698.832 + 1210.41i) q^{26} -729.000 q^{27} +(-1947.69 - 713.499i) q^{28} -759.033 q^{29} +(-1358.86 + 2353.62i) q^{30} +(-4515.87 - 7821.72i) q^{31} +(-512.000 - 886.810i) q^{32} +(-672.578 + 1164.94i) q^{33} -4599.39 q^{34} +(-1679.42 - 9641.80i) q^{35} +1296.00 q^{36} +(-3897.45 + 6750.57i) q^{37} +(5590.40 + 9682.86i) q^{38} +(1572.37 + 2723.43i) q^{39} +(2415.76 - 4184.21i) q^{40} +7640.49 q^{41} +(-3581.45 + 2992.51i) q^{42} +12188.8 q^{43} +(1195.69 - 2071.00i) q^{44} +(3057.44 + 5295.64i) q^{45} +(-3627.94 - 6283.77i) q^{46} +(-12299.4 + 21303.2i) q^{47} -2304.00 q^{48} +(2987.41 - 16539.4i) q^{49} +10296.4 q^{50} +(-5174.31 + 8962.17i) q^{51} +(-2795.33 - 4841.65i) q^{52} +(-6798.11 - 11774.7i) q^{53} +(1458.00 - 2525.33i) q^{54} +11283.2 q^{55} +(6367.02 - 5320.01i) q^{56} +25156.8 q^{57} +(1518.07 - 2629.37i) q^{58} +(13179.4 + 22827.4i) q^{59} +(-5435.45 - 9414.47i) q^{60} +(-17660.9 + 30589.5i) q^{61} +36127.0 q^{62} +(1801.94 + 10345.2i) q^{63} +4096.00 q^{64} +(13189.1 - 22844.2i) q^{65} +(-2690.31 - 4659.76i) q^{66} +(-27186.0 - 47087.5i) q^{67} +(9198.78 - 15932.7i) q^{68} -16325.7 q^{69} +(36759.0 + 13465.9i) q^{70} -70145.7 q^{71} +(-2592.00 + 4489.48i) q^{72} +(22234.4 + 38511.1i) q^{73} +(-15589.8 - 27002.3i) q^{74} +(11583.4 - 20063.1i) q^{75} -44723.2 q^{76} +(18194.1 + 6665.05i) q^{77} -12579.0 q^{78} +(-30806.2 + 53357.9i) q^{79} +(9663.02 + 16736.8i) q^{80} +(-3280.50 - 5681.99i) q^{81} +(-15281.0 + 26467.5i) q^{82} -87142.0 q^{83} +(-3203.45 - 18391.5i) q^{84} +86804.6 q^{85} +(-24377.7 + 42223.4i) q^{86} +(-3415.65 - 5916.08i) q^{87} +(4782.78 + 8284.01i) q^{88} +(-49284.7 + 85363.6i) q^{89} -24459.5 q^{90} +(34761.5 - 29045.3i) q^{91} +29023.5 q^{92} +(40642.9 - 70395.5i) q^{93} +(-49197.6 - 85212.8i) q^{94} +(-105508. - 182745. i) q^{95} +(4608.00 - 7981.29i) q^{96} +32342.3 q^{97} +(51319.2 + 43427.4i) q^{98} -12106.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} + 18 q^{3} - 32 q^{4} + 53 q^{5} - 144 q^{6} + 6 q^{7} + 256 q^{8} - 162 q^{9} + 212 q^{10} - 191 q^{11} + 288 q^{12} - 758 q^{13} - 36 q^{14} + 954 q^{15} - 512 q^{16} + 340 q^{17} - 648 q^{18}+ \cdots + 30942 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 + 3.46410i −0.353553 + 0.612372i
\(3\) 4.50000 + 7.79423i 0.288675 + 0.500000i
\(4\) −8.00000 − 13.8564i −0.250000 − 0.433013i
\(5\) 37.7462 − 65.3783i 0.675224 − 1.16952i −0.301179 − 0.953568i \(-0.597380\pi\)
0.976403 − 0.215955i \(-0.0692864\pi\)
\(6\) −36.0000 −0.408248
\(7\) 99.4847 − 83.1252i 0.767381 − 0.641191i
\(8\) 64.0000 0.353553
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 150.985 + 261.513i 0.477456 + 0.826977i
\(11\) 74.7309 + 129.438i 0.186217 + 0.322537i 0.943986 − 0.329986i \(-0.107044\pi\)
−0.757769 + 0.652523i \(0.773711\pi\)
\(12\) 72.0000 − 124.708i 0.144338 − 0.250000i
\(13\) 349.416 0.573435 0.286717 − 0.958015i \(-0.407436\pi\)
0.286717 + 0.958015i \(0.407436\pi\)
\(14\) 88.9847 + 510.875i 0.121338 + 0.696618i
\(15\) 679.431 0.779682
\(16\) −128.000 + 221.703i −0.125000 + 0.216506i
\(17\) 574.923 + 995.797i 0.482489 + 0.835696i 0.999798 − 0.0201029i \(-0.00639938\pi\)
−0.517309 + 0.855799i \(0.673066\pi\)
\(18\) −162.000 − 280.592i −0.117851 − 0.204124i
\(19\) 1397.60 − 2420.72i 0.888176 − 1.53837i 0.0461468 − 0.998935i \(-0.485306\pi\)
0.842029 − 0.539432i \(-0.181361\pi\)
\(20\) −1207.88 −0.675224
\(21\) 1095.58 + 401.343i 0.542119 + 0.198595i
\(22\) −597.847 −0.263350
\(23\) −906.985 + 1570.94i −0.357504 + 0.619214i −0.987543 − 0.157349i \(-0.949705\pi\)
0.630040 + 0.776563i \(0.283039\pi\)
\(24\) 288.000 + 498.831i 0.102062 + 0.176777i
\(25\) −1287.05 − 2229.23i −0.411855 − 0.713354i
\(26\) −698.832 + 1210.41i −0.202740 + 0.351156i
\(27\) −729.000 −0.192450
\(28\) −1947.69 − 713.499i −0.469489 − 0.171988i
\(29\) −759.033 −0.167597 −0.0837984 − 0.996483i \(-0.526705\pi\)
−0.0837984 + 0.996483i \(0.526705\pi\)
\(30\) −1358.86 + 2353.62i −0.275659 + 0.477456i
\(31\) −4515.87 − 7821.72i −0.843990 − 1.46183i −0.886495 − 0.462738i \(-0.846867\pi\)
0.0425050 − 0.999096i \(-0.486466\pi\)
\(32\) −512.000 − 886.810i −0.0883883 − 0.153093i
\(33\) −672.578 + 1164.94i −0.107512 + 0.186217i
\(34\) −4599.39 −0.682343
\(35\) −1679.42 − 9641.80i −0.231733 − 1.33042i
\(36\) 1296.00 0.166667
\(37\) −3897.45 + 6750.57i −0.468032 + 0.810655i −0.999333 − 0.0365280i \(-0.988370\pi\)
0.531300 + 0.847183i \(0.321704\pi\)
\(38\) 5590.40 + 9682.86i 0.628035 + 1.08779i
\(39\) 1572.37 + 2723.43i 0.165536 + 0.286717i
\(40\) 2415.76 − 4184.21i 0.238728 − 0.413489i
\(41\) 7640.49 0.709842 0.354921 − 0.934896i \(-0.384508\pi\)
0.354921 + 0.934896i \(0.384508\pi\)
\(42\) −3581.45 + 2992.51i −0.313282 + 0.261765i
\(43\) 12188.8 1.00529 0.502645 − 0.864493i \(-0.332360\pi\)
0.502645 + 0.864493i \(0.332360\pi\)
\(44\) 1195.69 − 2071.00i 0.0931083 − 0.161268i
\(45\) 3057.44 + 5295.64i 0.225075 + 0.389841i
\(46\) −3627.94 − 6283.77i −0.252793 − 0.437851i
\(47\) −12299.4 + 21303.2i −0.812156 + 1.40670i 0.0991964 + 0.995068i \(0.468373\pi\)
−0.911352 + 0.411627i \(0.864961\pi\)
\(48\) −2304.00 −0.144338
\(49\) 2987.41 − 16539.4i 0.177748 − 0.984076i
\(50\) 10296.4 0.582451
\(51\) −5174.31 + 8962.17i −0.278565 + 0.482489i
\(52\) −2795.33 − 4841.65i −0.143359 − 0.248305i
\(53\) −6798.11 − 11774.7i −0.332429 − 0.575783i 0.650559 − 0.759456i \(-0.274535\pi\)
−0.982988 + 0.183672i \(0.941201\pi\)
\(54\) 1458.00 − 2525.33i 0.0680414 − 0.117851i
\(55\) 11283.2 0.502952
\(56\) 6367.02 − 5320.01i 0.271310 − 0.226695i
\(57\) 25156.8 1.02558
\(58\) 1518.07 − 2629.37i 0.0592544 − 0.102632i
\(59\) 13179.4 + 22827.4i 0.492908 + 0.853742i 0.999967 − 0.00816991i \(-0.00260059\pi\)
−0.507059 + 0.861912i \(0.669267\pi\)
\(60\) −5435.45 − 9414.47i −0.194920 − 0.337612i
\(61\) −17660.9 + 30589.5i −0.607698 + 1.05256i 0.383921 + 0.923366i \(0.374573\pi\)
−0.991619 + 0.129198i \(0.958760\pi\)
\(62\) 36127.0 1.19358
\(63\) 1801.94 + 10345.2i 0.0571991 + 0.328389i
\(64\) 4096.00 0.125000
\(65\) 13189.1 − 22844.2i 0.387197 − 0.670645i
\(66\) −2690.31 − 4659.76i −0.0760226 − 0.131675i
\(67\) −27186.0 − 47087.5i −0.739874 − 1.28150i −0.952552 − 0.304377i \(-0.901552\pi\)
0.212678 − 0.977122i \(-0.431781\pi\)
\(68\) 9198.78 − 15932.7i 0.241245 − 0.417848i
\(69\) −16325.7 −0.412810
\(70\) 36759.0 + 13465.9i 0.896641 + 0.328467i
\(71\) −70145.7 −1.65141 −0.825706 − 0.564101i \(-0.809223\pi\)
−0.825706 + 0.564101i \(0.809223\pi\)
\(72\) −2592.00 + 4489.48i −0.0589256 + 0.102062i
\(73\) 22234.4 + 38511.1i 0.488335 + 0.845822i 0.999910 − 0.0134170i \(-0.00427090\pi\)
−0.511574 + 0.859239i \(0.670938\pi\)
\(74\) −15589.8 − 27002.3i −0.330949 − 0.573220i
\(75\) 11583.4 − 20063.1i 0.237785 − 0.411855i
\(76\) −44723.2 −0.888176
\(77\) 18194.1 + 6665.05i 0.349707 + 0.128108i
\(78\) −12579.0 −0.234104
\(79\) −30806.2 + 53357.9i −0.555355 + 0.961903i 0.442521 + 0.896758i \(0.354084\pi\)
−0.997876 + 0.0651450i \(0.979249\pi\)
\(80\) 9663.02 + 16736.8i 0.168806 + 0.292381i
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) −15281.0 + 26467.5i −0.250967 + 0.434688i
\(83\) −87142.0 −1.38846 −0.694228 − 0.719755i \(-0.744254\pi\)
−0.694228 + 0.719755i \(0.744254\pi\)
\(84\) −3203.45 − 18391.5i −0.0495358 − 0.284393i
\(85\) 86804.6 1.30315
\(86\) −24377.7 + 42223.4i −0.355423 + 0.615611i
\(87\) −3415.65 − 5916.08i −0.0483810 − 0.0837984i
\(88\) 4782.78 + 8284.01i 0.0658375 + 0.114034i
\(89\) −49284.7 + 85363.6i −0.659534 + 1.14235i 0.321203 + 0.947010i \(0.395913\pi\)
−0.980736 + 0.195336i \(0.937420\pi\)
\(90\) −24459.5 −0.318304
\(91\) 34761.5 − 29045.3i 0.440043 − 0.367681i
\(92\) 29023.5 0.357504
\(93\) 40642.9 − 70395.5i 0.487278 − 0.843990i
\(94\) −49197.6 − 85212.8i −0.574281 − 0.994684i
\(95\) −105508. − 182745.i −1.19944 − 2.07748i
\(96\) 4608.00 − 7981.29i 0.0510310 − 0.0883883i
\(97\) 32342.3 0.349013 0.174507 − 0.984656i \(-0.444167\pi\)
0.174507 + 0.984656i \(0.444167\pi\)
\(98\) 51319.2 + 43427.4i 0.539778 + 0.456771i
\(99\) −12106.4 −0.124144
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 42.6.e.c.37.2 yes 4
3.2 odd 2 126.6.g.h.37.1 4
4.3 odd 2 336.6.q.f.289.2 4
7.2 even 3 294.6.a.r.1.1 2
7.3 odd 6 294.6.e.s.67.1 4
7.4 even 3 inner 42.6.e.c.25.2 ✓ 4
7.5 odd 6 294.6.a.w.1.2 2
7.6 odd 2 294.6.e.s.79.1 4
21.2 odd 6 882.6.a.bh.1.2 2
21.5 even 6 882.6.a.bb.1.1 2
21.11 odd 6 126.6.g.h.109.1 4
28.11 odd 6 336.6.q.f.193.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.e.c.25.2 ✓ 4 7.4 even 3 inner
42.6.e.c.37.2 yes 4 1.1 even 1 trivial
126.6.g.h.37.1 4 3.2 odd 2
126.6.g.h.109.1 4 21.11 odd 6
294.6.a.r.1.1 2 7.2 even 3
294.6.a.w.1.2 2 7.5 odd 6
294.6.e.s.67.1 4 7.3 odd 6
294.6.e.s.79.1 4 7.6 odd 2
336.6.q.f.193.2 4 28.11 odd 6
336.6.q.f.289.2 4 4.3 odd 2
882.6.a.bb.1.1 2 21.5 even 6
882.6.a.bh.1.2 2 21.2 odd 6