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Properties

Label 21.4.c.b.20.1
Level $21$
Weight $4$
Character 21.20
Analytic conductor $1.239$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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(20,21)] 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("21.20"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] 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: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{-17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 46x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 20.1
Root \(6.57260i\) of defining polynomial
Character \(\chi\) \(=\) 21.20
Dual form 21.4.c.b.20.3

$q$-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-4.12311i q^{2} +(-5.04975 - 1.22474i) q^{3} -9.00000 q^{4} +10.0995 q^{5} +(-5.04975 + 20.8207i) q^{6} +(7.00000 - 17.1464i) q^{7} +4.12311i q^{8} +(24.0000 + 12.3693i) q^{9} -41.6413i q^{10} +32.9848i q^{11} +(45.4478 + 11.0227i) q^{12} +56.3383i q^{13} +(-70.6965 - 28.8617i) q^{14} +(-51.0000 - 12.3693i) q^{15} -55.0000 q^{16} +60.5970 q^{17} +(51.0000 - 98.9545i) q^{18} -36.7423i q^{19} -90.8955 q^{20} +(-56.3483 + 78.0120i) q^{21} +136.000 q^{22} -90.7083i q^{23} +(5.04975 - 20.8207i) q^{24} -23.0000 q^{25} +232.289 q^{26} +(-106.045 - 91.8559i) q^{27} +(-63.0000 + 154.318i) q^{28} +57.7235i q^{29} +(-51.0000 + 210.278i) q^{30} +254.747i q^{31} +259.756i q^{32} +(40.3980 - 166.565i) q^{33} -249.848i q^{34} +(70.6965 - 173.170i) q^{35} +(-216.000 - 111.324i) q^{36} +230.000 q^{37} -151.493 q^{38} +(69.0000 - 284.494i) q^{39} +41.6413i q^{40} -141.393 q^{41} +(321.652 + 232.330i) q^{42} +44.0000 q^{43} -296.864i q^{44} +(242.388 + 124.924i) q^{45} -374.000 q^{46} -343.383 q^{47} +(277.736 + 67.3610i) q^{48} +(-245.000 - 240.050i) q^{49} +94.8314i q^{50} +(-306.000 - 74.2159i) q^{51} -507.044i q^{52} +206.155i q^{53} +(-378.731 + 437.234i) q^{54} +333.131i q^{55} +(70.6965 + 28.8617i) q^{56} +(-45.0000 + 185.540i) q^{57} +238.000 q^{58} +131.294 q^{59} +(459.000 + 111.324i) q^{60} -71.0352i q^{61} +1050.35 q^{62} +(380.090 - 324.929i) q^{63} +631.000 q^{64} +568.989i q^{65} +(-686.766 - 166.565i) q^{66} -64.0000 q^{67} -545.373 q^{68} +(-111.095 + 458.055i) q^{69} +(-714.000 - 291.489i) q^{70} -461.788i q^{71} +(-51.0000 + 98.9545i) q^{72} -88.1816i q^{73} -948.314i q^{74} +(116.144 + 28.1691i) q^{75} +330.681i q^{76} +(565.572 + 230.894i) q^{77} +(-1173.00 - 284.494i) q^{78} -442.000 q^{79} -555.473 q^{80} +(423.000 + 593.727i) q^{81} +582.979i q^{82} -494.876 q^{83} +(507.134 - 702.108i) q^{84} +612.000 q^{85} -181.417i q^{86} +(70.6965 - 291.489i) q^{87} -136.000 q^{88} -484.776 q^{89} +(515.075 - 999.392i) q^{90} +(966.000 + 394.368i) q^{91} +816.375i q^{92} +(312.000 - 1286.41i) q^{93} +1415.81i q^{94} -371.080i q^{95} +(318.134 - 1311.70i) q^{96} -1092.47i q^{97} +(-989.751 + 1010.16i) q^{98} +(-408.000 + 791.636i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{4} + 28 q^{7} + 96 q^{9} - 204 q^{15} - 220 q^{16} + 204 q^{18} - 84 q^{21} + 544 q^{22} - 92 q^{25} - 252 q^{28} - 204 q^{30} - 864 q^{36} + 920 q^{37} + 276 q^{39} + 1428 q^{42} + 176 q^{43}+ \cdots - 1632 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.12311i − 1.45774i −0.684653 − 0.728869i \(-0.740046\pi\)
0.684653 − 0.728869i \(-0.259954\pi\)
\(3\) −5.04975 − 1.22474i −0.971825 − 0.235702i
\(4\) −9.00000 −1.12500
\(5\) 10.0995 0.903327 0.451664 − 0.892188i \(-0.350831\pi\)
0.451664 + 0.892188i \(0.350831\pi\)
\(6\) −5.04975 + 20.8207i −0.343592 + 1.41667i
\(7\) 7.00000 − 17.1464i 0.377964 − 0.925820i
\(8\) 4.12311i 0.182217i
\(9\) 24.0000 + 12.3693i 0.888889 + 0.458123i
\(10\) − 41.6413i − 1.31681i
\(11\) 32.9848i 0.904119i 0.891988 + 0.452059i \(0.149310\pi\)
−0.891988 + 0.452059i \(0.850690\pi\)
\(12\) 45.4478 + 11.0227i 1.09330 + 0.265165i
\(13\) 56.3383i 1.20196i 0.799266 + 0.600978i \(0.205222\pi\)
−0.799266 + 0.600978i \(0.794778\pi\)
\(14\) −70.6965 − 28.8617i −1.34960 − 0.550973i
\(15\) −51.0000 − 12.3693i −0.877876 − 0.212916i
\(16\) −55.0000 −0.859375
\(17\) 60.5970 0.864526 0.432263 − 0.901748i \(-0.357715\pi\)
0.432263 + 0.901748i \(0.357715\pi\)
\(18\) 51.0000 − 98.9545i 0.667823 − 1.29577i
\(19\) − 36.7423i − 0.443646i −0.975087 − 0.221823i \(-0.928799\pi\)
0.975087 − 0.221823i \(-0.0712007\pi\)
\(20\) −90.8955 −1.01624
\(21\) −56.3483 + 78.0120i −0.585533 + 0.810648i
\(22\) 136.000 1.31797
\(23\) − 90.7083i − 0.822348i −0.911557 − 0.411174i \(-0.865119\pi\)
0.911557 − 0.411174i \(-0.134881\pi\)
\(24\) 5.04975 − 20.8207i 0.0429490 − 0.177083i
\(25\) −23.0000 −0.184000
\(26\) 232.289 1.75214
\(27\) −106.045 − 91.8559i −0.755864 − 0.654729i
\(28\) −63.0000 + 154.318i −0.425210 + 1.04155i
\(29\) 57.7235i 0.369620i 0.982774 + 0.184810i \(0.0591670\pi\)
−0.982774 + 0.184810i \(0.940833\pi\)
\(30\) −51.0000 + 210.278i −0.310376 + 1.27971i
\(31\) 254.747i 1.47593i 0.674838 + 0.737966i \(0.264214\pi\)
−0.674838 + 0.737966i \(0.735786\pi\)
\(32\) 259.756i 1.43496i
\(33\) 40.3980 − 166.565i 0.213103 − 0.878645i
\(34\) − 249.848i − 1.26025i
\(35\) 70.6965 − 173.170i 0.341426 − 0.836318i
\(36\) −216.000 − 111.324i −1.00000 − 0.515388i
\(37\) 230.000 1.02194 0.510970 − 0.859599i \(-0.329286\pi\)
0.510970 + 0.859599i \(0.329286\pi\)
\(38\) −151.493 −0.646719
\(39\) 69.0000 − 284.494i 0.283304 − 1.16809i
\(40\) 41.6413i 0.164602i
\(41\) −141.393 −0.538583 −0.269291 − 0.963059i \(-0.586789\pi\)
−0.269291 + 0.963059i \(0.586789\pi\)
\(42\) 321.652 + 232.330i 1.18171 + 0.853554i
\(43\) 44.0000 0.156045 0.0780225 − 0.996952i \(-0.475139\pi\)
0.0780225 + 0.996952i \(0.475139\pi\)
\(44\) − 296.864i − 1.01713i
\(45\) 242.388 + 124.924i 0.802957 + 0.413835i
\(46\) −374.000 −1.19877
\(47\) −343.383 −1.06569 −0.532847 − 0.846212i \(-0.678878\pi\)
−0.532847 + 0.846212i \(0.678878\pi\)
\(48\) 277.736 + 67.3610i 0.835162 + 0.202557i
\(49\) −245.000 − 240.050i −0.714286 − 0.699854i
\(50\) 94.8314i 0.268224i
\(51\) −306.000 − 74.2159i −0.840168 − 0.203771i
\(52\) − 507.044i − 1.35220i
\(53\) 206.155i 0.534294i 0.963656 + 0.267147i \(0.0860810\pi\)
−0.963656 + 0.267147i \(0.913919\pi\)
\(54\) −378.731 + 437.234i −0.954423 + 1.10185i
\(55\) 333.131i 0.816715i
\(56\) 70.6965 + 28.8617i 0.168700 + 0.0688716i
\(57\) −45.0000 + 185.540i −0.104568 + 0.431146i
\(58\) 238.000 0.538809
\(59\) 131.294 0.289711 0.144856 − 0.989453i \(-0.453728\pi\)
0.144856 + 0.989453i \(0.453728\pi\)
\(60\) 459.000 + 111.324i 0.987611 + 0.239531i
\(61\) − 71.0352i − 0.149100i −0.997217 − 0.0745502i \(-0.976248\pi\)
0.997217 − 0.0745502i \(-0.0237521\pi\)
\(62\) 1050.35 2.15152
\(63\) 380.090 − 324.929i 0.760108 − 0.649797i
\(64\) 631.000 1.23242
\(65\) 568.989i 1.08576i
\(66\) −686.766 − 166.565i −1.28083 − 0.310648i
\(67\) −64.0000 −0.116699 −0.0583496 − 0.998296i \(-0.518584\pi\)
−0.0583496 + 0.998296i \(0.518584\pi\)
\(68\) −545.373 −0.972592
\(69\) −111.095 + 458.055i −0.193829 + 0.799178i
\(70\) −714.000 − 291.489i −1.21913 − 0.497709i
\(71\) − 461.788i − 0.771889i −0.922522 − 0.385945i \(-0.873876\pi\)
0.922522 − 0.385945i \(-0.126124\pi\)
\(72\) −51.0000 + 98.9545i −0.0834779 + 0.161971i
\(73\) − 88.1816i − 0.141382i −0.997498 − 0.0706910i \(-0.977480\pi\)
0.997498 − 0.0706910i \(-0.0225204\pi\)
\(74\) − 948.314i − 1.48972i
\(75\) 116.144 + 28.1691i 0.178816 + 0.0433692i
\(76\) 330.681i 0.499102i
\(77\) 565.572 + 230.894i 0.837051 + 0.341725i
\(78\) −1173.00 − 284.494i −1.70277 − 0.412982i
\(79\) −442.000 −0.629480 −0.314740 − 0.949178i \(-0.601917\pi\)
−0.314740 + 0.949178i \(0.601917\pi\)
\(80\) −555.473 −0.776297
\(81\) 423.000 + 593.727i 0.580247 + 0.814441i
\(82\) 582.979i 0.785112i
\(83\) −494.876 −0.654454 −0.327227 − 0.944946i \(-0.606114\pi\)
−0.327227 + 0.944946i \(0.606114\pi\)
\(84\) 507.134 − 702.108i 0.658725 − 0.911979i
\(85\) 612.000 0.780950
\(86\) − 181.417i − 0.227473i
\(87\) 70.6965 − 291.489i 0.0871203 − 0.359206i
\(88\) −136.000 −0.164746
\(89\) −484.776 −0.577373 −0.288686 − 0.957424i \(-0.593218\pi\)
−0.288686 + 0.957424i \(0.593218\pi\)
\(90\) 515.075 − 999.392i 0.603263 − 1.17050i
\(91\) 966.000 + 394.368i 1.11279 + 0.454297i
\(92\) 816.375i 0.925141i
\(93\) 312.000 − 1286.41i 0.347881 − 1.43435i
\(94\) 1415.81i 1.55350i
\(95\) − 371.080i − 0.400757i
\(96\) 318.134 − 1311.70i 0.338224 − 1.39453i
\(97\) − 1092.47i − 1.14354i −0.820413 − 0.571772i \(-0.806256\pi\)
0.820413 − 0.571772i \(-0.193744\pi\)
\(98\) −989.751 + 1010.16i −1.02020 + 1.04124i
\(99\) −408.000 + 791.636i −0.414197 + 0.803661i
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.c.b.20.1 ✓ 4
3.2 odd 2 inner 21.4.c.b.20.4 yes 4
4.3 odd 2 336.4.k.b.209.4 4
7.2 even 3 147.4.g.c.80.2 8
7.3 odd 6 147.4.g.c.68.3 8
7.4 even 3 147.4.g.c.68.4 8
7.5 odd 6 147.4.g.c.80.1 8
7.6 odd 2 inner 21.4.c.b.20.2 yes 4
12.11 even 2 336.4.k.b.209.2 4
21.2 odd 6 147.4.g.c.80.3 8
21.5 even 6 147.4.g.c.80.4 8
21.11 odd 6 147.4.g.c.68.1 8
21.17 even 6 147.4.g.c.68.2 8
21.20 even 2 inner 21.4.c.b.20.3 yes 4
28.27 even 2 336.4.k.b.209.1 4
84.83 odd 2 336.4.k.b.209.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.c.b.20.1 ✓ 4 1.1 even 1 trivial
21.4.c.b.20.2 yes 4 7.6 odd 2 inner
21.4.c.b.20.3 yes 4 21.20 even 2 inner
21.4.c.b.20.4 yes 4 3.2 odd 2 inner
147.4.g.c.68.1 8 21.11 odd 6
147.4.g.c.68.2 8 21.17 even 6
147.4.g.c.68.3 8 7.3 odd 6
147.4.g.c.68.4 8 7.4 even 3
147.4.g.c.80.1 8 7.5 odd 6
147.4.g.c.80.2 8 7.2 even 3
147.4.g.c.80.3 8 21.2 odd 6
147.4.g.c.80.4 8 21.5 even 6
336.4.k.b.209.1 4 28.27 even 2
336.4.k.b.209.2 4 12.11 even 2
336.4.k.b.209.3 4 84.83 odd 2
336.4.k.b.209.4 4 4.3 odd 2