↑

Properties

Label 21.8.e.b.4.2
Level $21$
Weight $8$
Character 21.4
Analytic conductor $6.560$
Analytic rank $0$
Dimension $10$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [21,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] 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: \(6.56008553517\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 412 x^{8} - 96 x^{7} + 133333 x^{6} - 66144 x^{5} + 15003636 x^{4} - 36459504 x^{3} + \cdots + 2149991424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 7^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 4.2
Root \(5.14307 + 8.90806i\) of defining polynomial
Character \(\chi\) \(=\) 21.4
Dual form 21.8.e.b.16.2

$q$-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+(-6.64307 - 11.5061i) q^{2} +(13.5000 - 23.3827i) q^{3} +(-24.2608 + 42.0210i) q^{4} +(-174.171 - 301.672i) q^{5} -358.726 q^{6} +(899.874 - 117.341i) q^{7} -1055.96 q^{8} +(-364.500 - 631.333i) q^{9} +(-2314.06 + 4008.06i) q^{10} +(-3804.31 + 6589.25i) q^{11} +(655.043 + 1134.57i) q^{12} -1072.14 q^{13} +(-7328.07 - 9574.58i) q^{14} -9405.21 q^{15} +(10120.2 + 17528.7i) q^{16} +(13747.0 - 23810.5i) q^{17} +(-4842.80 + 8387.98i) q^{18} +(-4796.31 - 8307.46i) q^{19} +16902.1 q^{20} +(9404.56 - 22625.6i) q^{21} +101089. q^{22} +(-48295.0 - 83649.3i) q^{23} +(-14255.5 + 24691.2i) q^{24} +(-21608.2 + 37426.6i) q^{25} +(7122.27 + 12336.1i) q^{26} -19683.0 q^{27} +(-16900.9 + 40660.4i) q^{28} +140816. q^{29} +(62479.5 + 108218. i) q^{30} +(-39856.4 + 69033.4i) q^{31} +(66877.1 - 115835. i) q^{32} +(102716. + 177910. i) q^{33} -365289. q^{34} +(-192130. - 251030. i) q^{35} +35372.3 q^{36} +(-260802. - 451723. i) q^{37} +(-63724.5 + 110374. i) q^{38} +(-14473.8 + 25069.4i) q^{39} +(183917. + 318554. i) q^{40} +117898. q^{41} +(-322808. + 42093.2i) q^{42} +235897. q^{43} +(-184591. - 319722. i) q^{44} +(-126970. + 219919. i) q^{45} +(-641654. + 1.11138e6i) q^{46} +(10902.1 + 18883.0i) q^{47} +546491. q^{48} +(796005. - 211184. i) q^{49} +574181. q^{50} +(-371169. - 642884. i) q^{51} +(26010.9 - 45052.2i) q^{52} +(146867. - 254380. i) q^{53} +(130756. + 226475. i) q^{54} +2.65039e6 q^{55} +(-950232. + 123907. i) q^{56} -259001. q^{57} +(-935451. - 1.62025e6i) q^{58} +(-333226. + 577164. i) q^{59} +(228178. - 395216. i) q^{60} +(-670187. - 1.16080e6i) q^{61} +1.05908e6 q^{62} +(-402085. - 525349. i) q^{63} +813695. q^{64} +(186734. + 323433. i) q^{65} +(1.36470e6 - 2.36374e6i) q^{66} +(-66617.5 + 115385. i) q^{67} +(667028. + 1.15533e6i) q^{68} -2.60793e6 q^{69} +(-1.61205e6 + 3.87829e6i) q^{70} -1.66963e6 q^{71} +(384898. + 666662. i) q^{72} +(165691. - 286985. i) q^{73} +(-3.46506e6 + 6.00166e6i) q^{74} +(583423. + 1.01052e6i) q^{75} +465450. q^{76} +(-2.65021e6 + 6.37590e6i) q^{77} +384603. q^{78} +(-2.73564e6 - 4.73826e6i) q^{79} +(3.52528e6 - 6.10597e6i) q^{80} +(-265720. + 460241. i) q^{81} +(-783208. - 1.35656e6i) q^{82} +5.14452e6 q^{83} +(722588. + 944105. i) q^{84} -9.57729e6 q^{85} +(-1.56708e6 - 2.71426e6i) q^{86} +(1.90102e6 - 3.29266e6i) q^{87} +(4.01720e6 - 6.95799e6i) q^{88} +(2.40006e6 + 4.15702e6i) q^{89} +3.37389e6 q^{90} +(-964787. + 125805. i) q^{91} +4.68671e6 q^{92} +(1.07612e6 + 1.86390e6i) q^{93} +(144847. - 250883. i) q^{94} +(-1.67075e6 + 2.89383e6i) q^{95} +(-1.80568e6 - 3.12753e6i) q^{96} +1.42272e7 q^{97} +(-7.71784e6 - 7.75604e6i) q^{98} +5.54668e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 15 q^{2} + 135 q^{3} - 229 q^{4} + 198 q^{5} - 810 q^{6} - 859 q^{7} + 7134 q^{8} - 3645 q^{9} + 5081 q^{10} - 7248 q^{11} + 6183 q^{12} - 2546 q^{13} - 54 q^{14} + 10692 q^{15} + 14759 q^{16} + 34764 q^{17}+ \cdots + 10567584 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.64307 − 11.5061i −0.587170 − 1.01701i −0.994601 − 0.103773i \(-0.966909\pi\)
0.407431 − 0.913236i \(-0.366425\pi\)
\(3\) 13.5000 − 23.3827i 0.288675 − 0.500000i
\(4\) −24.2608 + 42.0210i −0.189538 + 0.328289i
\(5\) −174.171 − 301.672i −0.623131 − 1.07930i −0.988899 − 0.148589i \(-0.952527\pi\)
0.365768 − 0.930706i \(-0.380806\pi\)
\(6\) −358.726 −0.678006
\(7\) 899.874 − 117.341i 0.991605 − 0.129302i
\(8\) −1055.96 −0.729177
\(9\) −364.500 − 631.333i −0.166667 − 0.288675i
\(10\) −2314.06 + 4008.06i −0.731768 + 1.26746i
\(11\) −3804.31 + 6589.25i −0.861789 + 1.49266i 0.00841110 + 0.999965i \(0.497323\pi\)
−0.870200 + 0.492698i \(0.836011\pi\)
\(12\) 655.043 + 1134.57i 0.109430 + 0.189538i
\(13\) −1072.14 −0.135347 −0.0676734 − 0.997708i \(-0.521558\pi\)
−0.0676734 + 0.997708i \(0.521558\pi\)
\(14\) −7328.07 − 9574.58i −0.713743 − 0.932549i
\(15\) −9405.21 −0.719530
\(16\) 10120.2 + 17528.7i 0.617689 + 1.06987i
\(17\) 13747.0 − 23810.5i 0.678636 − 1.17543i −0.296756 − 0.954953i \(-0.595905\pi\)
0.975392 − 0.220479i \(-0.0707619\pi\)
\(18\) −4842.80 + 8387.98i −0.195723 + 0.339003i
\(19\) −4796.31 − 8307.46i −0.160424 − 0.277863i 0.774597 − 0.632456i \(-0.217953\pi\)
−0.935021 + 0.354593i \(0.884620\pi\)
\(20\) 16902.1 0.472428
\(21\) 9404.56 − 22625.6i 0.221601 − 0.533129i
\(22\) 101089. 2.02407
\(23\) −48295.0 − 83649.3i −0.827665 − 1.43356i −0.899866 − 0.436167i \(-0.856336\pi\)
0.0722008 − 0.997390i \(-0.476998\pi\)
\(24\) −14255.5 + 24691.2i −0.210495 + 0.364588i
\(25\) −21608.2 + 37426.6i −0.276586 + 0.479060i
\(26\) 7122.27 + 12336.1i 0.0794716 + 0.137649i
\(27\) −19683.0 −0.192450
\(28\) −16900.9 + 40660.4i −0.145498 + 0.350041i
\(29\) 140816. 1.07216 0.536079 − 0.844168i \(-0.319905\pi\)
0.536079 + 0.844168i \(0.319905\pi\)
\(30\) 62479.5 + 108218.i 0.422487 + 0.731768i
\(31\) −39856.4 + 69033.4i −0.240288 + 0.416191i −0.960796 − 0.277255i \(-0.910575\pi\)
0.720508 + 0.693446i \(0.243909\pi\)
\(32\) 66877.1 − 115835.i 0.360789 − 0.624904i
\(33\) 102716. + 177910.i 0.497554 + 0.861789i
\(34\) −365289. −1.59390
\(35\) −192130. − 251030.i −0.757456 − 0.989662i
\(36\) 35372.3 0.126359
\(37\) −260802. − 451723.i −0.846458 − 1.46611i −0.884349 − 0.466827i \(-0.845397\pi\)
0.0378902 − 0.999282i \(-0.487936\pi\)
\(38\) −63724.5 + 110374.i −0.188393 + 0.326306i
\(39\) −14473.8 + 25069.4i −0.0390712 + 0.0676734i
\(40\) 183917. + 318554.i 0.454373 + 0.786997i
\(41\) 117898. 0.267156 0.133578 − 0.991038i \(-0.457353\pi\)
0.133578 + 0.991038i \(0.457353\pi\)
\(42\) −322808. + 42093.2i −0.672314 + 0.0876678i
\(43\) 235897. 0.452462 0.226231 − 0.974074i \(-0.427360\pi\)
0.226231 + 0.974074i \(0.427360\pi\)
\(44\) −184591. − 319722.i −0.326683 − 0.565832i
\(45\) −126970. + 219919.i −0.207710 + 0.359765i
\(46\) −641654. + 1.11138e6i −0.971960 + 1.68348i
\(47\) 10902.1 + 18883.0i 0.0153168 + 0.0265295i 0.873582 − 0.486677i \(-0.161791\pi\)
−0.858265 + 0.513206i \(0.828458\pi\)
\(48\) 546491. 0.713245
\(49\) 796005. − 211184.i 0.966562 − 0.256434i
\(50\) 574181. 0.649611
\(51\) −371169. − 642884.i −0.391811 − 0.678636i
\(52\) 26010.9 − 45052.2i 0.0256533 − 0.0444329i
\(53\) 146867. − 254380.i 0.135506 − 0.234703i −0.790285 − 0.612740i \(-0.790067\pi\)
0.925790 + 0.378037i \(0.123401\pi\)
\(54\) 130756. + 226475.i 0.113001 + 0.195723i
\(55\) 2.65039e6 2.14803
\(56\) −950232. + 123907.i −0.723055 + 0.0942842i
\(57\) −259001. −0.185242
\(58\) −935451. − 1.62025e6i −0.629540 − 1.09040i
\(59\) −333226. + 577164.i −0.211231 + 0.365862i −0.952100 − 0.305787i \(-0.901080\pi\)
0.740869 + 0.671649i \(0.234414\pi\)
\(60\) 228178. − 395216.i 0.136378 − 0.236214i
\(61\) −670187. − 1.16080e6i −0.378043 − 0.654790i 0.612734 − 0.790289i \(-0.290070\pi\)
−0.990777 + 0.135499i \(0.956736\pi\)
\(62\) 1.05908e6 0.564360
\(63\) −402085. − 525349.i −0.202594 − 0.264701i
\(64\) 813695. 0.388000
\(65\) 186734. + 323433.i 0.0843388 + 0.146079i
\(66\) 1.36470e6 − 2.36374e6i 0.584298 − 1.01203i
\(67\) −66617.5 + 115385.i −0.0270599 + 0.0468692i −0.879238 − 0.476382i \(-0.841948\pi\)
0.852178 + 0.523251i \(0.175281\pi\)
\(68\) 667028. + 1.15533e6i 0.257254 + 0.445578i
\(69\) −2.60793e6 −0.955705
\(70\) −1.61205e6 + 3.87829e6i −0.561740 + 1.35144i
\(71\) −1.66963e6 −0.553627 −0.276813 − 0.960924i \(-0.589278\pi\)
−0.276813 + 0.960924i \(0.589278\pi\)
\(72\) 384898. + 666662.i 0.121529 + 0.210495i
\(73\) 165691. − 286985.i 0.0498504 − 0.0863434i −0.840023 − 0.542550i \(-0.817459\pi\)
0.889874 + 0.456207i \(0.150792\pi\)
\(74\) −3.46506e6 + 6.00166e6i −0.994030 + 1.72171i
\(75\) 583423. + 1.01052e6i 0.159687 + 0.276586i
\(76\) 465450. 0.121626
\(77\) −2.65021e6 + 6.37590e6i −0.661550 + 1.59156i
\(78\) 384603. 0.0917659
\(79\) −2.73564e6 − 4.73826e6i −0.624257 − 1.08124i −0.988684 − 0.150013i \(-0.952068\pi\)
0.364427 − 0.931232i \(-0.381265\pi\)
\(80\) 3.52528e6 − 6.10597e6i 0.769802 − 1.33334i
\(81\) −265720. + 460241.i −0.0555556 + 0.0962250i
\(82\) −783208. − 1.35656e6i −0.156866 − 0.271700i
\(83\) 5.14452e6 0.987578 0.493789 − 0.869582i \(-0.335612\pi\)
0.493789 + 0.869582i \(0.335612\pi\)
\(84\) 722588. + 944105.i 0.133019 + 0.173797i
\(85\) −9.57729e6 −1.69152
\(86\) −1.56708e6 − 2.71426e6i −0.265672 − 0.460158i
\(87\) 1.90102e6 − 3.29266e6i 0.309506 − 0.536079i
\(88\) 4.01720e6 − 6.95799e6i 0.628396 − 1.08841i
\(89\) 2.40006e6 + 4.15702e6i 0.360875 + 0.625054i 0.988105 − 0.153780i \(-0.0491448\pi\)
−0.627230 + 0.778834i \(0.715811\pi\)
\(90\) 3.37389e6 0.487846
\(91\) −964787. + 125805.i −0.134211 + 0.0175007i
\(92\) 4.68671e6 0.627495
\(93\) 1.07612e6 + 1.86390e6i 0.138730 + 0.240288i
\(94\) 144847. − 250883.i 0.0179872 − 0.0311547i
\(95\) −1.67075e6 + 2.89383e6i −0.199931 + 0.346290i
\(96\) −1.80568e6 − 3.12753e6i −0.208301 − 0.360789i
\(97\) 1.42272e7 1.58277 0.791387 − 0.611315i \(-0.209359\pi\)
0.791387 + 0.611315i \(0.209359\pi\)
\(98\) −7.71784e6 − 7.75604e6i −0.828332 − 0.832432i
\(99\) 5.54668e6 0.574526
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 21.8.e.b.4.2 ✓ 10
3.2 odd 2 63.8.e.d.46.4 10
7.2 even 3 inner 21.8.e.b.16.2 yes 10
7.3 odd 6 147.8.a.k.1.4 5
7.4 even 3 147.8.a.j.1.4 5
7.5 odd 6 147.8.e.n.79.2 10
7.6 odd 2 147.8.e.n.67.2 10
21.2 odd 6 63.8.e.d.37.4 10
21.11 odd 6 441.8.a.x.1.2 5
21.17 even 6 441.8.a.w.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.8.e.b.4.2 ✓ 10 1.1 even 1 trivial
21.8.e.b.16.2 yes 10 7.2 even 3 inner
63.8.e.d.37.4 10 21.2 odd 6
63.8.e.d.46.4 10 3.2 odd 2
147.8.a.j.1.4 5 7.4 even 3
147.8.a.k.1.4 5 7.3 odd 6
147.8.e.n.67.2 10 7.6 odd 2
147.8.e.n.79.2 10 7.5 odd 6
441.8.a.w.1.2 5 21.17 even 6
441.8.a.x.1.2 5 21.11 odd 6