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Properties

Label 21.4.g.a.5.6
Level $21$
Weight $4$
Character 21.5
Analytic conductor $1.239$
Analytic rank $0$
Dimension $12$
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(5,21)] 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("21.5"); 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([3, 5])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 29x^{9} + 6x^{8} - 49x^{7} + 1564x^{6} - 441x^{5} + 486x^{4} - 21141x^{3} - 59049x + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 5.6
Root \(-0.232749 + 2.99096i\) of defining polynomial
Character \(\chi\) \(=\) 21.5
Dual form 21.4.g.a.17.6

$q$-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+(3.93653 + 2.27276i) q^{2} +(-5.18049 - 0.403134i) q^{3} +(6.33084 + 10.9653i) q^{4} +(5.80193 - 10.0492i) q^{5} +(-19.4769 - 13.3609i) q^{6} +(-18.4018 - 2.09174i) q^{7} +21.1897i q^{8} +(26.6750 + 4.17686i) q^{9} +(45.6790 - 26.3728i) q^{10} +(-15.5157 + 8.95800i) q^{11} +(-28.3764 - 59.3580i) q^{12} +62.4185i q^{13} +(-67.6850 - 50.0569i) q^{14} +(-34.1081 + 49.7211i) q^{15} +(2.48762 - 4.30868i) q^{16} +(-10.7082 - 18.5472i) q^{17} +(95.5138 + 77.0680i) q^{18} +(9.50747 + 5.48914i) q^{19} +146.925 q^{20} +(94.4869 + 18.2546i) q^{21} -81.4374 q^{22} +(59.8367 + 34.5467i) q^{23} +(8.54230 - 109.773i) q^{24} +(-4.82490 - 8.35697i) q^{25} +(-141.862 + 245.712i) q^{26} +(-136.506 - 32.3918i) q^{27} +(-93.5619 - 215.024i) q^{28} -265.583i q^{29} +(-247.271 + 118.209i) q^{30} +(8.85795 - 5.11414i) q^{31} +(166.392 - 96.0665i) q^{32} +(83.9902 - 40.1519i) q^{33} -97.3486i q^{34} +(-127.786 + 172.788i) q^{35} +(123.074 + 318.943i) q^{36} +(-20.8257 + 36.0712i) q^{37} +(24.9510 + 43.2163i) q^{38} +(25.1630 - 323.358i) q^{39} +(212.941 + 122.941i) q^{40} +31.0035 q^{41} +(330.462 + 286.606i) q^{42} -224.550 q^{43} +(-196.455 - 113.423i) q^{44} +(196.741 - 243.829i) q^{45} +(157.033 + 271.988i) q^{46} +(81.8595 - 141.785i) q^{47} +(-14.6241 + 21.3182i) q^{48} +(334.249 + 76.9836i) q^{49} -43.8633i q^{50} +(47.9968 + 100.400i) q^{51} +(-684.440 + 395.161i) q^{52} +(-456.586 + 263.610i) q^{53} +(-463.740 - 437.755i) q^{54} +207.895i q^{55} +(44.3235 - 389.928i) q^{56} +(-47.0405 - 32.2692i) q^{57} +(603.606 - 1045.48i) q^{58} +(205.978 + 356.765i) q^{59} +(-761.141 - 59.2302i) q^{60} +(223.807 + 129.215i) q^{61} +46.4928 q^{62} +(-482.129 - 132.659i) q^{63} +833.541 q^{64} +(627.258 + 362.148i) q^{65} +(421.886 + 32.8302i) q^{66} +(-161.737 - 280.137i) q^{67} +(135.584 - 234.838i) q^{68} +(-296.056 - 203.091i) q^{69} +(-895.738 + 389.756i) q^{70} +45.4199i q^{71} +(-88.5066 + 565.236i) q^{72} +(-486.879 + 281.100i) q^{73} +(-163.962 + 94.6635i) q^{74} +(21.6264 + 45.2383i) q^{75} +139.004i q^{76} +(304.254 - 132.388i) q^{77} +(833.969 - 1215.72i) q^{78} +(-144.610 + 250.473i) q^{79} +(-28.8660 - 49.9974i) q^{80} +(694.108 + 222.835i) q^{81} +(122.046 + 70.4635i) q^{82} -448.767 q^{83} +(398.013 + 1151.65i) q^{84} -248.513 q^{85} +(-883.949 - 510.348i) q^{86} +(-107.066 + 1375.85i) q^{87} +(-189.818 - 328.774i) q^{88} +(280.814 - 486.384i) q^{89} +(1328.64 - 512.698i) q^{90} +(130.563 - 1148.61i) q^{91} +874.839i q^{92} +(-47.9502 + 22.9228i) q^{93} +(644.484 - 372.093i) q^{94} +(110.323 - 63.6953i) q^{95} +(-900.720 + 430.593i) q^{96} -214.364i q^{97} +(1140.82 + 1062.71i) q^{98} +(-451.297 + 174.147i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} + 14 q^{4} - 56 q^{7} - 3 q^{9} + 30 q^{10} - 192 q^{12} + 6 q^{15} + 134 q^{16} + 66 q^{18} + 300 q^{19} + 357 q^{21} - 268 q^{22} + 414 q^{24} - 42 q^{25} - 602 q^{28} - 822 q^{30} - 930 q^{31}+ \cdots - 3354 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.93653 + 2.27276i 1.39177 + 0.803541i 0.993512 − 0.113731i \(-0.0362803\pi\)
0.398262 + 0.917272i \(0.369614\pi\)
\(3\) −5.18049 − 0.403134i −0.996986 − 0.0775831i
\(4\) 6.33084 + 10.9653i 0.791355 + 1.37067i
\(5\) 5.80193 − 10.0492i 0.518941 − 0.898832i −0.480817 − 0.876821i \(-0.659660\pi\)
0.999758 − 0.0220109i \(-0.00700684\pi\)
\(6\) −19.4769 − 13.3609i −1.32524 − 0.909097i
\(7\) −18.4018 − 2.09174i −0.993601 − 0.112944i
\(8\) 21.1897i 0.936463i
\(9\) 26.6750 + 4.17686i 0.987962 + 0.154699i
\(10\) 45.6790 − 26.3728i 1.44450 − 0.833980i
\(11\) −15.5157 + 8.95800i −0.425287 + 0.245540i −0.697337 − 0.716743i \(-0.745632\pi\)
0.272050 + 0.962283i \(0.412299\pi\)
\(12\) −28.3764 − 59.3580i −0.682629 − 1.42793i
\(13\) 62.4185i 1.33167i 0.746097 + 0.665837i \(0.231925\pi\)
−0.746097 + 0.665837i \(0.768075\pi\)
\(14\) −67.6850 − 50.0569i −1.29211 − 0.955591i
\(15\) −34.1081 + 49.7211i −0.587111 + 0.855862i
\(16\) 2.48762 − 4.30868i 0.0388690 − 0.0673231i
\(17\) −10.7082 − 18.5472i −0.152772 − 0.264609i 0.779474 − 0.626435i \(-0.215487\pi\)
−0.932245 + 0.361826i \(0.882153\pi\)
\(18\) 95.5138 + 77.0680i 1.25071 + 1.00917i
\(19\) 9.50747 + 5.48914i 0.114798 + 0.0662787i 0.556300 − 0.830982i \(-0.312221\pi\)
−0.441502 + 0.897261i \(0.645554\pi\)
\(20\) 146.925 1.64267
\(21\) 94.4869 + 18.2546i 0.981844 + 0.189690i
\(22\) −81.4374 −0.789205
\(23\) 59.8367 + 34.5467i 0.542470 + 0.313195i 0.746079 − 0.665857i \(-0.231934\pi\)
−0.203609 + 0.979052i \(0.565267\pi\)
\(24\) 8.54230 − 109.773i 0.0726537 − 0.933641i
\(25\) −4.82490 − 8.35697i −0.0385992 − 0.0668557i
\(26\) −141.862 + 245.712i −1.07005 + 1.85339i
\(27\) −136.506 − 32.3918i −0.972982 − 0.230881i
\(28\) −93.5619 − 215.024i −0.631484 − 1.45128i
\(29\) − 265.583i − 1.70061i −0.526294 − 0.850303i \(-0.676419\pi\)
0.526294 − 0.850303i \(-0.323581\pi\)
\(30\) −247.271 + 118.209i −1.50484 + 0.719398i
\(31\) 8.85795 − 5.11414i 0.0513205 − 0.0296299i −0.474120 − 0.880460i \(-0.657234\pi\)
0.525441 + 0.850830i \(0.323900\pi\)
\(32\) 166.392 − 96.0665i 0.919195 − 0.530697i
\(33\) 83.9902 − 40.1519i 0.443055 − 0.211805i
\(34\) − 97.3486i − 0.491034i
\(35\) −127.786 + 172.788i −0.617138 + 0.834470i
\(36\) 123.074 + 318.943i 0.569788 + 1.47659i
\(37\) −20.8257 + 36.0712i −0.0925331 + 0.160272i −0.908576 − 0.417719i \(-0.862830\pi\)
0.816043 + 0.577991i \(0.196163\pi\)
\(38\) 24.9510 + 43.2163i 0.106515 + 0.184490i
\(39\) 25.1630 − 323.358i 0.103315 − 1.32766i
\(40\) 212.941 + 122.941i 0.841723 + 0.485969i
\(41\) 31.0035 0.118096 0.0590480 − 0.998255i \(-0.481193\pi\)
0.0590480 + 0.998255i \(0.481193\pi\)
\(42\) 330.462 + 286.606i 1.21408 + 1.05296i
\(43\) −224.550 −0.796363 −0.398181 − 0.917307i \(-0.630359\pi\)
−0.398181 + 0.917307i \(0.630359\pi\)
\(44\) −196.455 − 113.423i −0.673107 − 0.388618i
\(45\) 196.741 − 243.829i 0.651742 − 0.807732i
\(46\) 157.033 + 271.988i 0.503330 + 0.871793i
\(47\) 81.8595 − 141.785i 0.254052 − 0.440031i −0.710586 − 0.703611i \(-0.751570\pi\)
0.964638 + 0.263580i \(0.0849033\pi\)
\(48\) −14.6241 + 21.3182i −0.0439750 + 0.0641046i
\(49\) 334.249 + 76.9836i 0.974487 + 0.224442i
\(50\) − 43.8633i − 0.124064i
\(51\) 47.9968 + 100.400i 0.131782 + 0.275664i
\(52\) −684.440 + 395.161i −1.82528 + 1.05383i
\(53\) −456.586 + 263.610i −1.18334 + 0.683200i −0.956784 − 0.290799i \(-0.906079\pi\)
−0.226553 + 0.973999i \(0.572746\pi\)
\(54\) −463.740 − 437.755i −1.16865 − 1.10317i
\(55\) 207.895i 0.509682i
\(56\) 44.3235 − 389.928i 0.105768 − 0.930471i
\(57\) −47.0405 − 32.2692i −0.109310 − 0.0749853i
\(58\) 603.606 − 1045.48i 1.36651 − 2.36686i
\(59\) 205.978 + 356.765i 0.454510 + 0.787234i 0.998660 − 0.0517537i \(-0.0164811\pi\)
−0.544150 + 0.838988i \(0.683148\pi\)
\(60\) −761.141 − 59.2302i −1.63771 − 0.127443i
\(61\) 223.807 + 129.215i 0.469764 + 0.271218i 0.716141 − 0.697956i \(-0.245907\pi\)
−0.246377 + 0.969174i \(0.579240\pi\)
\(62\) 46.4928 0.0952352
\(63\) −482.129 − 132.659i −0.964168 − 0.265293i
\(64\) 833.541 1.62801
\(65\) 627.258 + 362.148i 1.19695 + 0.691060i
\(66\) 421.886 + 32.8302i 0.786826 + 0.0612290i
\(67\) −161.737 − 280.137i −0.294915 − 0.510808i 0.680050 − 0.733166i \(-0.261958\pi\)
−0.974965 + 0.222357i \(0.928625\pi\)
\(68\) 135.584 − 234.838i 0.241794 − 0.418799i
\(69\) −296.056 − 203.091i −0.516536 − 0.354338i
\(70\) −895.738 + 389.756i −1.52945 + 0.665497i
\(71\) 45.4199i 0.0759205i 0.999279 + 0.0379603i \(0.0120860\pi\)
−0.999279 + 0.0379603i \(0.987914\pi\)
\(72\) −88.5066 + 565.236i −0.144869 + 0.925190i
\(73\) −486.879 + 281.100i −0.780615 + 0.450688i −0.836648 − 0.547741i \(-0.815488\pi\)
0.0560334 + 0.998429i \(0.482155\pi\)
\(74\) −163.962 + 94.6635i −0.257570 + 0.148708i
\(75\) 21.6264 + 45.2383i 0.0332960 + 0.0696489i
\(76\) 139.004i 0.209800i
\(77\) 304.254 − 132.388i 0.450298 − 0.195935i
\(78\) 833.969 − 1215.72i 1.21062 − 1.76478i
\(79\) −144.610 + 250.473i −0.205949 + 0.356714i −0.950435 − 0.310925i \(-0.899361\pi\)
0.744486 + 0.667638i \(0.232695\pi\)
\(80\) −28.8660 − 49.9974i −0.0403415 − 0.0698735i
\(81\) 694.108 + 222.835i 0.952137 + 0.305673i
\(82\) 122.046 + 70.4635i 0.164363 + 0.0948950i
\(83\) −448.767 −0.593477 −0.296738 − 0.954959i \(-0.595899\pi\)
−0.296738 + 0.954959i \(0.595899\pi\)
\(84\) 398.013 + 1151.65i 0.516986 + 1.49589i
\(85\) −248.513 −0.317118
\(86\) −883.949 − 510.348i −1.10836 − 0.639910i
\(87\) −107.066 + 1375.85i −0.131938 + 1.69548i
\(88\) −189.818 − 328.774i −0.229939 − 0.398266i
\(89\) 280.814 − 486.384i 0.334452 − 0.579288i −0.648927 − 0.760850i \(-0.724782\pi\)
0.983379 + 0.181562i \(0.0581155\pi\)
\(90\) 1328.64 − 512.698i 1.55612 − 0.600479i
\(91\) 130.563 − 1148.61i 0.150404 − 1.32315i
\(92\) 874.839i 0.991394i
\(93\) −47.9502 + 22.9228i −0.0534645 + 0.0255590i
\(94\) 644.484 − 372.093i 0.707165 − 0.408282i
\(95\) 110.323 − 63.6953i 0.119147 − 0.0687895i
\(96\) −900.720 + 430.593i −0.957597 + 0.457784i
\(97\) − 214.364i − 0.224385i −0.993686 − 0.112192i \(-0.964213\pi\)
0.993686 − 0.112192i \(-0.0357873\pi\)
\(98\) 1140.82 + 1062.71i 1.17592 + 1.09541i
\(99\) −451.297 + 174.147i −0.458152 + 0.176793i
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.g.a.5.6 yes 12
3.2 odd 2 inner 21.4.g.a.5.1 ✓ 12
4.3 odd 2 336.4.bc.d.257.6 12
7.2 even 3 147.4.c.a.146.2 12
7.3 odd 6 inner 21.4.g.a.17.1 yes 12
7.4 even 3 147.4.g.d.80.1 12
7.5 odd 6 147.4.c.a.146.1 12
7.6 odd 2 147.4.g.d.68.6 12
12.11 even 2 336.4.bc.d.257.4 12
21.2 odd 6 147.4.c.a.146.11 12
21.5 even 6 147.4.c.a.146.12 12
21.11 odd 6 147.4.g.d.80.6 12
21.17 even 6 inner 21.4.g.a.17.6 yes 12
21.20 even 2 147.4.g.d.68.1 12
28.3 even 6 336.4.bc.d.17.4 12
84.59 odd 6 336.4.bc.d.17.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.g.a.5.1 ✓ 12 3.2 odd 2 inner
21.4.g.a.5.6 yes 12 1.1 even 1 trivial
21.4.g.a.17.1 yes 12 7.3 odd 6 inner
21.4.g.a.17.6 yes 12 21.17 even 6 inner
147.4.c.a.146.1 12 7.5 odd 6
147.4.c.a.146.2 12 7.2 even 3
147.4.c.a.146.11 12 21.2 odd 6
147.4.c.a.146.12 12 21.5 even 6
147.4.g.d.68.1 12 21.20 even 2
147.4.g.d.68.6 12 7.6 odd 2
147.4.g.d.80.1 12 7.4 even 3
147.4.g.d.80.6 12 21.11 odd 6
336.4.bc.d.17.4 12 28.3 even 6
336.4.bc.d.17.6 12 84.59 odd 6
336.4.bc.d.257.4 12 12.11 even 2
336.4.bc.d.257.6 12 4.3 odd 2