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Properties

Label 21.4.e.a.4.1
Level $21$
Weight $4$
Character 21.4
Analytic conductor $1.239$
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] = [21,4,Mod(4,21)] 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("21.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.23904011012\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 4.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.4
Dual form 21.4.e.a.16.1

$q$-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+(1.50000 + 2.59808i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(1.50000 + 2.59808i) q^{5} -9.00000 q^{6} +(-3.50000 - 18.1865i) q^{7} +21.0000 q^{8} +(-4.50000 - 7.79423i) q^{9} +(-4.50000 + 7.79423i) q^{10} +(7.50000 - 12.9904i) q^{11} +(-1.50000 - 2.59808i) q^{12} -64.0000 q^{13} +(42.0000 - 36.3731i) q^{14} -9.00000 q^{15} +(35.5000 + 61.4878i) q^{16} +(-42.0000 + 72.7461i) q^{17} +(13.5000 - 23.3827i) q^{18} +(8.00000 + 13.8564i) q^{19} -3.00000 q^{20} +(52.5000 + 18.1865i) q^{21} +45.0000 q^{22} +(42.0000 + 72.7461i) q^{23} +(-31.5000 + 54.5596i) q^{24} +(58.0000 - 100.459i) q^{25} +(-96.0000 - 166.277i) q^{26} +27.0000 q^{27} +(17.5000 + 6.06218i) q^{28} -297.000 q^{29} +(-13.5000 - 23.3827i) q^{30} +(126.500 - 219.104i) q^{31} +(-22.5000 + 38.9711i) q^{32} +(22.5000 + 38.9711i) q^{33} -252.000 q^{34} +(42.0000 - 36.3731i) q^{35} +9.00000 q^{36} +(158.000 + 273.664i) q^{37} +(-24.0000 + 41.5692i) q^{38} +(96.0000 - 166.277i) q^{39} +(31.5000 + 54.5596i) q^{40} +360.000 q^{41} +(31.5000 + 163.679i) q^{42} +26.0000 q^{43} +(7.50000 + 12.9904i) q^{44} +(13.5000 - 23.3827i) q^{45} +(-126.000 + 218.238i) q^{46} +(15.0000 + 25.9808i) q^{47} -213.000 q^{48} +(-318.500 + 127.306i) q^{49} +348.000 q^{50} +(-126.000 - 218.238i) q^{51} +(32.0000 - 55.4256i) q^{52} +(-181.500 + 314.367i) q^{53} +(40.5000 + 70.1481i) q^{54} +45.0000 q^{55} +(-73.5000 - 381.917i) q^{56} -48.0000 q^{57} +(-445.500 - 771.629i) q^{58} +(7.50000 - 12.9904i) q^{59} +(4.50000 - 7.79423i) q^{60} +(59.0000 + 102.191i) q^{61} +759.000 q^{62} +(-126.000 + 109.119i) q^{63} +433.000 q^{64} +(-96.0000 - 166.277i) q^{65} +(-67.5000 + 116.913i) q^{66} +(185.000 - 320.429i) q^{67} +(-42.0000 - 72.7461i) q^{68} -252.000 q^{69} +(157.500 + 54.5596i) q^{70} -342.000 q^{71} +(-94.5000 - 163.679i) q^{72} +(-181.000 + 313.501i) q^{73} +(-474.000 + 820.992i) q^{74} +(174.000 + 301.377i) q^{75} -16.0000 q^{76} +(-262.500 - 90.9327i) q^{77} +576.000 q^{78} +(-233.500 - 404.434i) q^{79} +(-106.500 + 184.463i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(540.000 + 935.307i) q^{82} +477.000 q^{83} +(-42.0000 + 36.3731i) q^{84} -252.000 q^{85} +(39.0000 + 67.5500i) q^{86} +(445.500 - 771.629i) q^{87} +(157.500 - 272.798i) q^{88} +(-453.000 - 784.619i) q^{89} +81.0000 q^{90} +(224.000 + 1163.94i) q^{91} -84.0000 q^{92} +(379.500 + 657.313i) q^{93} +(-45.0000 + 77.9423i) q^{94} +(-24.0000 + 41.5692i) q^{95} +(-67.5000 - 116.913i) q^{96} +503.000 q^{97} +(-808.500 - 636.529i) q^{98} -135.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 3 q^{3} - q^{4} + 3 q^{5} - 18 q^{6} - 7 q^{7} + 42 q^{8} - 9 q^{9} - 9 q^{10} + 15 q^{11} - 3 q^{12} - 128 q^{13} + 84 q^{14} - 18 q^{15} + 71 q^{16} - 84 q^{17} + 27 q^{18} + 16 q^{19}+ \cdots - 270 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.50000 + 2.59808i 0.530330 + 0.918559i 0.999374 + 0.0353837i \(0.0112653\pi\)
−0.469044 + 0.883175i \(0.655401\pi\)
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(5\) 1.50000 + 2.59808i 0.134164 + 0.232379i 0.925278 − 0.379290i \(-0.123832\pi\)
−0.791114 + 0.611669i \(0.790498\pi\)
\(6\) −9.00000 −0.612372
\(7\) −3.50000 − 18.1865i −0.188982 − 0.981981i
\(8\) 21.0000 0.928078
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) −4.50000 + 7.79423i −0.142302 + 0.246475i
\(11\) 7.50000 − 12.9904i 0.205576 − 0.356068i −0.744740 − 0.667355i \(-0.767427\pi\)
0.950316 + 0.311287i \(0.100760\pi\)
\(12\) −1.50000 − 2.59808i −0.0360844 − 0.0625000i
\(13\) −64.0000 −1.36542 −0.682708 − 0.730691i \(-0.739198\pi\)
−0.682708 + 0.730691i \(0.739198\pi\)
\(14\) 42.0000 − 36.3731i 0.801784 − 0.694365i
\(15\) −9.00000 −0.154919
\(16\) 35.5000 + 61.4878i 0.554688 + 0.960747i
\(17\) −42.0000 + 72.7461i −0.599206 + 1.03785i 0.393733 + 0.919225i \(0.371183\pi\)
−0.992939 + 0.118630i \(0.962150\pi\)
\(18\) 13.5000 − 23.3827i 0.176777 − 0.306186i
\(19\) 8.00000 + 13.8564i 0.0965961 + 0.167309i 0.910274 − 0.414007i \(-0.135871\pi\)
−0.813678 + 0.581317i \(0.802538\pi\)
\(20\) −3.00000 −0.0335410
\(21\) 52.5000 + 18.1865i 0.545545 + 0.188982i
\(22\) 45.0000 0.436092
\(23\) 42.0000 + 72.7461i 0.380765 + 0.659505i 0.991172 − 0.132583i \(-0.0423272\pi\)
−0.610406 + 0.792088i \(0.708994\pi\)
\(24\) −31.5000 + 54.5596i −0.267913 + 0.464039i
\(25\) 58.0000 − 100.459i 0.464000 − 0.803672i
\(26\) −96.0000 − 166.277i −0.724121 − 1.25421i
\(27\) 27.0000 0.192450
\(28\) 17.5000 + 6.06218i 0.118114 + 0.0409159i
\(29\) −297.000 −1.90178 −0.950888 − 0.309535i \(-0.899827\pi\)
−0.950888 + 0.309535i \(0.899827\pi\)
\(30\) −13.5000 − 23.3827i −0.0821584 − 0.142302i
\(31\) 126.500 − 219.104i 0.732906 − 1.26943i −0.222731 − 0.974880i \(-0.571497\pi\)
0.955636 − 0.294550i \(-0.0951696\pi\)
\(32\) −22.5000 + 38.9711i −0.124296 + 0.215287i
\(33\) 22.5000 + 38.9711i 0.118689 + 0.205576i
\(34\) −252.000 −1.27111
\(35\) 42.0000 − 36.3731i 0.202837 − 0.175662i
\(36\) 9.00000 0.0416667
\(37\) 158.000 + 273.664i 0.702028 + 1.21595i 0.967753 + 0.251900i \(0.0810553\pi\)
−0.265725 + 0.964049i \(0.585611\pi\)
\(38\) −24.0000 + 41.5692i −0.102456 + 0.177458i
\(39\) 96.0000 − 166.277i 0.394162 − 0.682708i
\(40\) 31.5000 + 54.5596i 0.124515 + 0.215666i
\(41\) 360.000 1.37128 0.685641 − 0.727940i \(-0.259522\pi\)
0.685641 + 0.727940i \(0.259522\pi\)
\(42\) 31.5000 + 163.679i 0.115728 + 0.601338i
\(43\) 26.0000 0.0922084 0.0461042 − 0.998937i \(-0.485319\pi\)
0.0461042 + 0.998937i \(0.485319\pi\)
\(44\) 7.50000 + 12.9904i 0.0256970 + 0.0445085i
\(45\) 13.5000 − 23.3827i 0.0447214 − 0.0774597i
\(46\) −126.000 + 218.238i −0.403863 + 0.699511i
\(47\) 15.0000 + 25.9808i 0.0465527 + 0.0806316i 0.888363 − 0.459142i \(-0.151843\pi\)
−0.841810 + 0.539774i \(0.818510\pi\)
\(48\) −213.000 −0.640498
\(49\) −318.500 + 127.306i −0.928571 + 0.371154i
\(50\) 348.000 0.984293
\(51\) −126.000 − 218.238i −0.345952 − 0.599206i
\(52\) 32.0000 − 55.4256i 0.0853385 − 0.147811i
\(53\) −181.500 + 314.367i −0.470395 + 0.814748i −0.999427 − 0.0338538i \(-0.989222\pi\)
0.529032 + 0.848602i \(0.322555\pi\)
\(54\) 40.5000 + 70.1481i 0.102062 + 0.176777i
\(55\) 45.0000 0.110324
\(56\) −73.5000 − 381.917i −0.175390 − 0.911354i
\(57\) −48.0000 −0.111540
\(58\) −445.500 − 771.629i −1.00857 − 1.74689i
\(59\) 7.50000 − 12.9904i 0.0165494 − 0.0286645i −0.857632 − 0.514264i \(-0.828065\pi\)
0.874182 + 0.485599i \(0.161399\pi\)
\(60\) 4.50000 − 7.79423i 0.00968246 − 0.0167705i
\(61\) 59.0000 + 102.191i 0.123839 + 0.214495i 0.921279 − 0.388903i \(-0.127146\pi\)
−0.797440 + 0.603399i \(0.793813\pi\)
\(62\) 759.000 1.55473
\(63\) −126.000 + 109.119i −0.251976 + 0.218218i
\(64\) 433.000 0.845703
\(65\) −96.0000 − 166.277i −0.183190 − 0.317294i
\(66\) −67.5000 + 116.913i −0.125889 + 0.218046i
\(67\) 185.000 − 320.429i 0.337334 − 0.584279i −0.646597 − 0.762832i \(-0.723808\pi\)
0.983930 + 0.178553i \(0.0571417\pi\)
\(68\) −42.0000 − 72.7461i −0.0749007 − 0.129732i
\(69\) −252.000 −0.439670
\(70\) 157.500 + 54.5596i 0.268926 + 0.0931589i
\(71\) −342.000 −0.571661 −0.285831 − 0.958280i \(-0.592269\pi\)
−0.285831 + 0.958280i \(0.592269\pi\)
\(72\) −94.5000 − 163.679i −0.154680 − 0.267913i
\(73\) −181.000 + 313.501i −0.290198 + 0.502638i −0.973856 − 0.227165i \(-0.927054\pi\)
0.683658 + 0.729802i \(0.260388\pi\)
\(74\) −474.000 + 820.992i −0.744613 + 1.28971i
\(75\) 174.000 + 301.377i 0.267891 + 0.464000i
\(76\) −16.0000 −0.0241490
\(77\) −262.500 − 90.9327i −0.388502 − 0.134581i
\(78\) 576.000 0.836143
\(79\) −233.500 − 404.434i −0.332542 − 0.575979i 0.650468 − 0.759534i \(-0.274573\pi\)
−0.983010 + 0.183555i \(0.941240\pi\)
\(80\) −106.500 + 184.463i −0.148838 + 0.257795i
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 540.000 + 935.307i 0.727232 + 1.25960i
\(83\) 477.000 0.630814 0.315407 − 0.948957i \(-0.397859\pi\)
0.315407 + 0.948957i \(0.397859\pi\)
\(84\) −42.0000 + 36.3731i −0.0545545 + 0.0472456i
\(85\) −252.000 −0.321568
\(86\) 39.0000 + 67.5500i 0.0489009 + 0.0846989i
\(87\) 445.500 − 771.629i 0.548996 − 0.950888i
\(88\) 157.500 − 272.798i 0.190790 − 0.330459i
\(89\) −453.000 − 784.619i −0.539527 − 0.934488i −0.998929 − 0.0462600i \(-0.985270\pi\)
0.459402 − 0.888228i \(-0.348064\pi\)
\(90\) 81.0000 0.0948683
\(91\) 224.000 + 1163.94i 0.258039 + 1.34081i
\(92\) −84.0000 −0.0951914
\(93\) 379.500 + 657.313i 0.423143 + 0.732906i
\(94\) −45.0000 + 77.9423i −0.0493765 + 0.0855227i
\(95\) −24.0000 + 41.5692i −0.0259195 + 0.0448938i
\(96\) −67.5000 − 116.913i −0.0717624 − 0.124296i
\(97\) 503.000 0.526515 0.263257 − 0.964726i \(-0.415203\pi\)
0.263257 + 0.964726i \(0.415203\pi\)
\(98\) −808.500 − 636.529i −0.833376 − 0.656113i
\(99\) −135.000 −0.137051
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 21.4.e.a.4.1 ✓ 2
3.2 odd 2 63.4.e.a.46.1 2
4.3 odd 2 336.4.q.e.193.1 2
7.2 even 3 inner 21.4.e.a.16.1 yes 2
7.3 odd 6 147.4.a.a.1.1 1
7.4 even 3 147.4.a.b.1.1 1
7.5 odd 6 147.4.e.h.79.1 2
7.6 odd 2 147.4.e.h.67.1 2
21.2 odd 6 63.4.e.a.37.1 2
21.5 even 6 441.4.e.c.226.1 2
21.11 odd 6 441.4.a.l.1.1 1
21.17 even 6 441.4.a.k.1.1 1
21.20 even 2 441.4.e.c.361.1 2
28.3 even 6 2352.4.a.bd.1.1 1
28.11 odd 6 2352.4.a.i.1.1 1
28.23 odd 6 336.4.q.e.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.a.4.1 ✓ 2 1.1 even 1 trivial
21.4.e.a.16.1 yes 2 7.2 even 3 inner
63.4.e.a.37.1 2 21.2 odd 6
63.4.e.a.46.1 2 3.2 odd 2
147.4.a.a.1.1 1 7.3 odd 6
147.4.a.b.1.1 1 7.4 even 3
147.4.e.h.67.1 2 7.6 odd 2
147.4.e.h.79.1 2 7.5 odd 6
336.4.q.e.193.1 2 4.3 odd 2
336.4.q.e.289.1 2 28.23 odd 6
441.4.a.k.1.1 1 21.17 even 6
441.4.a.l.1.1 1 21.11 odd 6
441.4.e.c.226.1 2 21.5 even 6
441.4.e.c.361.1 2 21.20 even 2
2352.4.a.i.1.1 1 28.11 odd 6
2352.4.a.bd.1.1 1 28.3 even 6