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Properties

Label 21.4.g.a.5.1
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 5.1
Root \(2.70662 + 1.29391i\) of defining polynomial
Character \(\chi\) \(=\) 21.5
Dual form 21.4.g.a.17.1

$q$-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+(-3.93653 - 2.27276i) q^{2} +(-2.24112 + 4.68800i) 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} +(-16.9548 - 21.0128i) q^{9} +(45.6790 - 26.3728i) q^{10} +(15.5157 - 8.95800i) q^{11} +(-65.5937 + 5.10435i) 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} +(18.9860 + 121.251i) q^{18} +(9.50747 + 5.48914i) q^{19} -146.925 q^{20} +(51.0467 - 81.5796i) q^{21} -81.4374 q^{22} +(-59.8367 - 34.5467i) q^{23} +(99.3376 + 47.4888i) 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} +(21.2635 + 273.248i) q^{30} +(8.85795 - 5.11414i) q^{31} +(-166.392 + 96.0665i) q^{32} +(7.22254 + 92.8136i) 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} +(-292.618 - 139.887i) q^{39} +(212.941 + 122.941i) q^{40} -31.0035 q^{41} +(-386.357 + 205.124i) q^{42} -224.550 q^{43} +(196.455 + 113.423i) q^{44} +(309.533 - 48.4678i) 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} +(-110.948 + 8.63368i) q^{51} +(-684.440 + 395.161i) q^{52} +(456.586 - 263.610i) q^{53} +(-610.977 - 182.733i) 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} +(329.276 - 688.783i) q^{60} +(223.807 + 129.215i) q^{61} -46.4928 q^{62} +(268.044 + 422.137i) q^{63} +833.541 q^{64} +(-627.258 - 362.148i) q^{65} +(182.511 - 381.779i) 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} +(-445.255 + 359.267i) q^{72} +(-486.879 + 281.100i) q^{73} +(163.962 - 94.6635i) q^{74} +(49.9907 - 3.89016i) 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} +(-154.073 + 712.532i) q^{81} +(122.046 + 70.4635i) q^{82} +448.767 q^{83} +(1217.72 + 43.2763i) q^{84} -248.513 q^{85} +(883.949 + 510.348i) q^{86} +(-1245.05 - 595.204i) 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} +(4.12336 + 52.9875i) q^{93} +(644.484 - 372.093i) q^{94} +(-110.323 + 63.6953i) q^{95} +(-77.4553 - 995.343i) 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 −0.398262 − 0.917272i \(-0.630386\pi\)
−0.993512 + 0.113731i \(0.963720\pi\)
\(3\) −2.24112 + 4.68800i −0.431304 + 0.902207i
\(4\) 6.33084 + 10.9653i 0.791355 + 1.37067i
\(5\) −5.80193 + 10.0492i −0.518941 + 0.898832i 0.480817 + 0.876821i \(0.340340\pi\)
−0.999758 + 0.0220109i \(0.992993\pi\)
\(6\) 19.4769 − 13.3609i 1.32524 − 0.909097i
\(7\) −18.4018 − 2.09174i −0.993601 − 0.112944i
\(8\) − 21.1897i − 0.936463i
\(9\) −16.9548 − 21.0128i −0.627954 − 0.778251i
\(10\) 45.6790 − 26.3728i 1.44450 − 0.833980i
\(11\) 15.5157 − 8.95800i 0.425287 − 0.245540i −0.272050 − 0.962283i \(-0.587701\pi\)
0.697337 + 0.716743i \(0.254368\pi\)
\(12\) −65.5937 + 5.10435i −1.57794 + 0.122792i
\(13\) 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 0.932245 − 0.361826i \(-0.117847\pi\)
−0.779474 + 0.626435i \(0.784513\pi\)
\(18\) 18.9860 + 121.251i 0.248613 + 1.58773i
\(19\) 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
\(21\) 51.0467 − 81.5796i 0.530443 − 0.847721i
\(22\) −81.4374 −0.789205
\(23\) −59.8367 − 34.5467i −0.542470 − 0.313195i 0.203609 − 0.979052i \(-0.434733\pi\)
−0.746079 + 0.665857i \(0.768066\pi\)
\(24\) 99.3376 + 47.4888i 0.844883 + 0.403900i
\(25\) −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 \(0.323581\pi\)
−0.526294 + 0.850303i \(0.676419\pi\)
\(30\) 21.2635 + 273.248i 0.129406 + 1.66293i
\(31\) 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
\(33\) 7.22254 + 92.8136i 0.0380995 + 0.489599i
\(34\) − 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
\(39\) −292.618 − 139.887i −1.20145 − 0.574356i
\(40\) 212.941 + 122.941i 0.841723 + 0.485969i
\(41\) −31.0035 −0.118096 −0.0590480 − 0.998255i \(-0.518807\pi\)
−0.0590480 + 0.998255i \(0.518807\pi\)
\(42\) −386.357 + 205.124i −1.41943 + 0.753603i
\(43\) −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
\(45\) 309.533 − 48.4678i 1.02539 − 0.160559i
\(46\) 157.033 + 271.988i 0.503330 + 0.871793i
\(47\) −81.8595 + 141.785i −0.254052 + 0.440031i −0.964638 − 0.263580i \(-0.915097\pi\)
0.710586 + 0.703611i \(0.248430\pi\)
\(48\) 14.6241 + 21.3182i 0.0439750 + 0.0641046i
\(49\) 334.249 + 76.9836i 0.974487 + 0.224442i
\(50\) 43.8633i 0.124064i
\(51\) −110.948 + 8.63368i −0.304623 + 0.0237050i
\(52\) −684.440 + 395.161i −1.82528 + 1.05383i
\(53\) 456.586 − 263.610i 1.18334 − 0.683200i 0.226553 − 0.973999i \(-0.427254\pi\)
0.956784 + 0.290799i \(0.0939211\pi\)
\(54\) −610.977 − 182.733i −1.53969 − 0.460496i
\(55\) 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 0.544150 − 0.838988i \(-0.316852\pi\)
−0.998660 + 0.0517537i \(0.983519\pi\)
\(60\) 329.276 − 688.783i 0.708488 − 1.48202i
\(61\) 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
\(63\) 268.044 + 422.137i 0.536037 + 0.844194i
\(64\) 833.541 1.62801
\(65\) −627.258 − 362.148i −1.19695 − 0.691060i
\(66\) 182.511 − 381.779i 0.340387 − 0.712026i
\(67\) −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 \(-0.987914\pi\)
0.999279 − 0.0379603i \(-0.0120860\pi\)
\(72\) −445.255 + 359.267i −0.728803 + 0.588056i
\(73\) −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
\(75\) 49.9907 − 3.89016i 0.0769657 − 0.00598929i
\(76\) 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
\(81\) −154.073 + 712.532i −0.211348 + 0.977411i
\(82\) 122.046 + 70.4635i 0.164363 + 0.0948950i
\(83\) 448.767 0.593477 0.296738 − 0.954959i \(-0.404101\pi\)
0.296738 + 0.954959i \(0.404101\pi\)
\(84\) 1217.72 + 43.2763i 1.58171 + 0.0562123i
\(85\) −248.513 −0.317118
\(86\) 883.949 + 510.348i 1.10836 + 0.639910i
\(87\) −1245.05 − 595.204i −1.53430 − 0.733478i
\(88\) −189.818 − 328.774i −0.229939 − 0.398266i
\(89\) −280.814 + 486.384i −0.334452 + 0.579288i −0.983379 − 0.181562i \(-0.941885\pi\)
0.648927 + 0.760850i \(0.275218\pi\)
\(90\) −1328.64 − 512.698i −1.55612 − 0.600479i
\(91\) 130.563 − 1148.61i 0.150404 − 1.32315i
\(92\) − 874.839i − 0.991394i
\(93\) 4.12336 + 52.9875i 0.00459756 + 0.0590811i
\(94\) 644.484 − 372.093i 0.707165 − 0.408282i
\(95\) −110.323 + 63.6953i −0.119147 + 0.0687895i
\(96\) −77.4553 − 995.343i −0.0823463 − 1.05820i
\(97\) − 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.5.1 ✓ 12
3.2 odd 2 inner 21.4.g.a.5.6 yes 12
4.3 odd 2 336.4.bc.d.257.4 12
7.2 even 3 147.4.c.a.146.11 12
7.3 odd 6 inner 21.4.g.a.17.6 yes 12
7.4 even 3 147.4.g.d.80.6 12
7.5 odd 6 147.4.c.a.146.12 12
7.6 odd 2 147.4.g.d.68.1 12
12.11 even 2 336.4.bc.d.257.6 12
21.2 odd 6 147.4.c.a.146.2 12
21.5 even 6 147.4.c.a.146.1 12
21.11 odd 6 147.4.g.d.80.1 12
21.17 even 6 inner 21.4.g.a.17.1 yes 12
21.20 even 2 147.4.g.d.68.6 12
28.3 even 6 336.4.bc.d.17.6 12
84.59 odd 6 336.4.bc.d.17.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.g.a.5.1 ✓ 12 1.1 even 1 trivial
21.4.g.a.5.6 yes 12 3.2 odd 2 inner
21.4.g.a.17.1 yes 12 21.17 even 6 inner
21.4.g.a.17.6 yes 12 7.3 odd 6 inner
147.4.c.a.146.1 12 21.5 even 6
147.4.c.a.146.2 12 21.2 odd 6
147.4.c.a.146.11 12 7.2 even 3
147.4.c.a.146.12 12 7.5 odd 6
147.4.g.d.68.1 12 7.6 odd 2
147.4.g.d.68.6 12 21.20 even 2
147.4.g.d.80.1 12 21.11 odd 6
147.4.g.d.80.6 12 7.4 even 3
336.4.bc.d.17.4 12 84.59 odd 6
336.4.bc.d.17.6 12 28.3 even 6
336.4.bc.d.257.4 12 4.3 odd 2
336.4.bc.d.257.6 12 12.11 even 2