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Properties

Label 21.3.h.b.2.1
Level $21$
Weight $3$
Character 21.2
Analytic conductor $0.572$
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] = [21,3,Mod(2,21)] 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("21.2"); S:= CuspForms(chi, 3); 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, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), 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: \(0.572208555157\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 2.1
Root \(-1.93649 - 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 21.2
Dual form 21.3.h.b.11.1

$q$-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+(-1.93649 - 1.11803i) q^{2} +(0.936492 - 2.85008i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.93649 + 1.11803i) q^{5} +(-5.00000 + 4.47214i) q^{6} +(3.50000 + 6.06218i) q^{7} +6.70820i q^{8} +(-7.24597 - 5.33816i) q^{9} +(-2.50000 - 4.33013i) q^{10} +(9.68246 - 5.59017i) q^{11} +(2.93649 - 0.614017i) q^{12} -2.00000 q^{13} -15.6525i q^{14} +(5.00000 - 4.47214i) q^{15} +(9.50000 - 16.4545i) q^{16} +(-23.2379 + 13.4164i) q^{17} +(8.06351 + 18.4385i) q^{18} +(-8.00000 + 13.8564i) q^{19} +2.23607i q^{20} +(20.5554 - 4.29812i) q^{21} -25.0000 q^{22} +(11.6190 + 6.70820i) q^{23} +(19.1190 + 6.28218i) q^{24} +(-10.0000 - 17.3205i) q^{25} +(3.87298 + 2.23607i) q^{26} +(-22.0000 + 15.6525i) q^{27} +(-3.50000 + 6.06218i) q^{28} -15.6525i q^{29} +(-14.6825 + 3.07008i) q^{30} +(1.50000 + 2.59808i) q^{31} +(-13.5554 + 7.82624i) q^{32} +(-6.86492 - 32.8310i) q^{33} +60.0000 q^{34} +15.6525i q^{35} +(1.00000 - 8.94427i) q^{36} +(-6.00000 + 10.3923i) q^{37} +(30.9839 - 17.8885i) q^{38} +(-1.87298 + 5.70017i) q^{39} +(-7.50000 + 12.9904i) q^{40} -31.3050i q^{41} +(-44.6109 - 14.6584i) q^{42} +44.0000 q^{43} +(9.68246 + 5.59017i) q^{44} +(-8.06351 - 18.4385i) q^{45} +(-15.0000 - 25.9808i) q^{46} +(-11.6190 - 6.70820i) q^{47} +(-38.0000 - 42.4853i) q^{48} +(-24.5000 + 42.4352i) q^{49} +44.7214i q^{50} +(16.4758 + 78.7943i) q^{51} +(-1.00000 - 1.73205i) q^{52} +(-17.4284 + 10.0623i) q^{53} +(60.1028 - 5.71414i) q^{54} +25.0000 q^{55} +(-40.6663 + 23.4787i) q^{56} +(32.0000 + 35.7771i) q^{57} +(-17.5000 + 30.3109i) q^{58} +(17.4284 - 10.0623i) q^{59} +(6.37298 + 2.09406i) q^{60} +(13.0000 - 22.5167i) q^{61} -6.70820i q^{62} +(7.00000 - 62.6099i) q^{63} -41.0000 q^{64} +(-3.87298 - 2.23607i) q^{65} +(-23.4123 + 71.2521i) q^{66} +(-26.0000 - 45.0333i) q^{67} +(-23.2379 - 13.4164i) q^{68} +(30.0000 - 26.8328i) q^{69} +(17.5000 - 30.3109i) q^{70} -93.9149i q^{71} +(35.8095 - 48.6074i) q^{72} +(-9.00000 - 15.5885i) q^{73} +(23.2379 - 13.4164i) q^{74} +(-58.7298 + 12.2803i) q^{75} -16.0000 q^{76} +(67.7772 + 39.1312i) q^{77} +(10.0000 - 8.94427i) q^{78} +(39.5000 - 68.4160i) q^{79} +(36.7933 - 21.2426i) q^{80} +(24.0081 + 77.3603i) q^{81} +(-35.0000 + 60.6218i) q^{82} +140.872i q^{83} +(14.0000 + 15.6525i) q^{84} -60.0000 q^{85} +(-85.2056 - 49.1935i) q^{86} +(-44.6109 - 14.6584i) q^{87} +(37.5000 + 64.9519i) q^{88} +(42.6028 + 24.5967i) q^{89} +(-5.00000 + 44.7214i) q^{90} +(-7.00000 - 12.1244i) q^{91} +13.4164i q^{92} +(8.80948 - 1.84205i) q^{93} +(15.0000 + 25.9808i) q^{94} +(-30.9839 + 17.8885i) q^{95} +(9.61088 + 45.9634i) q^{96} -93.0000 q^{97} +(94.8881 - 54.7837i) q^{98} +(-100.000 - 11.1803i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 2 q^{4} - 20 q^{6} + 14 q^{7} + 2 q^{9} - 10 q^{10} + 4 q^{12} - 8 q^{13} + 20 q^{15} + 38 q^{16} + 40 q^{18} - 32 q^{19} + 28 q^{21} - 100 q^{22} + 30 q^{24} - 40 q^{25} - 88 q^{27} - 14 q^{28}+ \cdots - 400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.93649 − 1.11803i −0.968246 − 0.559017i −0.0695448 − 0.997579i \(-0.522155\pi\)
−0.898701 + 0.438562i \(0.855488\pi\)
\(3\) 0.936492 − 2.85008i 0.312164 − 0.950028i
\(4\) 0.500000 + 0.866025i 0.125000 + 0.216506i
\(5\) 1.93649 + 1.11803i 0.387298 + 0.223607i 0.680989 − 0.732294i \(-0.261550\pi\)
−0.293691 + 0.955901i \(0.594884\pi\)
\(6\) −5.00000 + 4.47214i −0.833333 + 0.745356i
\(7\) 3.50000 + 6.06218i 0.500000 + 0.866025i
\(8\) 6.70820i 0.838525i
\(9\) −7.24597 − 5.33816i −0.805107 − 0.593129i
\(10\) −2.50000 − 4.33013i −0.250000 − 0.433013i
\(11\) 9.68246 − 5.59017i 0.880223 − 0.508197i 0.00949140 − 0.999955i \(-0.496979\pi\)
0.870732 + 0.491758i \(0.163645\pi\)
\(12\) 2.93649 − 0.614017i 0.244708 − 0.0511681i
\(13\) −2.00000 −0.153846 −0.0769231 − 0.997037i \(-0.524510\pi\)
−0.0769231 + 0.997037i \(0.524510\pi\)
\(14\) − 15.6525i − 1.11803i
\(15\) 5.00000 − 4.47214i 0.333333 − 0.298142i
\(16\) 9.50000 − 16.4545i 0.593750 − 1.02841i
\(17\) −23.2379 + 13.4164i −1.36694 + 0.789200i −0.990535 − 0.137257i \(-0.956171\pi\)
−0.376400 + 0.926457i \(0.622838\pi\)
\(18\) 8.06351 + 18.4385i 0.447973 + 1.02436i
\(19\) −8.00000 + 13.8564i −0.421053 + 0.729285i −0.996043 − 0.0888758i \(-0.971673\pi\)
0.574990 + 0.818160i \(0.305006\pi\)
\(20\) 2.23607i 0.111803i
\(21\) 20.5554 − 4.29812i 0.978831 − 0.204672i
\(22\) −25.0000 −1.13636
\(23\) 11.6190 + 6.70820i 0.505172 + 0.291661i 0.730847 − 0.682542i \(-0.239125\pi\)
−0.225675 + 0.974203i \(0.572459\pi\)
\(24\) 19.1190 + 6.28218i 0.796623 + 0.261757i
\(25\) −10.0000 − 17.3205i −0.400000 − 0.692820i
\(26\) 3.87298 + 2.23607i 0.148961 + 0.0860026i
\(27\) −22.0000 + 15.6525i −0.814815 + 0.579721i
\(28\) −3.50000 + 6.06218i −0.125000 + 0.216506i
\(29\) − 15.6525i − 0.539741i −0.962897 − 0.269870i \(-0.913019\pi\)
0.962897 − 0.269870i \(-0.0869808\pi\)
\(30\) −14.6825 + 3.07008i −0.489415 + 0.102336i
\(31\) 1.50000 + 2.59808i 0.0483871 + 0.0838089i 0.889205 − 0.457510i \(-0.151259\pi\)
−0.840817 + 0.541319i \(0.817925\pi\)
\(32\) −13.5554 + 7.82624i −0.423608 + 0.244570i
\(33\) −6.86492 − 32.8310i −0.208028 − 0.994878i
\(34\) 60.0000 1.76471
\(35\) 15.6525i 0.447214i
\(36\) 1.00000 − 8.94427i 0.0277778 − 0.248452i
\(37\) −6.00000 + 10.3923i −0.162162 + 0.280873i −0.935644 − 0.352946i \(-0.885180\pi\)
0.773482 + 0.633819i \(0.218513\pi\)
\(38\) 30.9839 − 17.8885i 0.815365 − 0.470751i
\(39\) −1.87298 + 5.70017i −0.0480252 + 0.146158i
\(40\) −7.50000 + 12.9904i −0.187500 + 0.324760i
\(41\) − 31.3050i − 0.763535i −0.924258 − 0.381768i \(-0.875315\pi\)
0.924258 − 0.381768i \(-0.124685\pi\)
\(42\) −44.6109 − 14.6584i −1.06216 − 0.349010i
\(43\) 44.0000 1.02326 0.511628 − 0.859207i \(-0.329043\pi\)
0.511628 + 0.859207i \(0.329043\pi\)
\(44\) 9.68246 + 5.59017i 0.220056 + 0.127049i
\(45\) −8.06351 − 18.4385i −0.179189 − 0.409745i
\(46\) −15.0000 − 25.9808i −0.326087 − 0.564799i
\(47\) −11.6190 − 6.70820i −0.247212 − 0.142728i 0.371275 − 0.928523i \(-0.378921\pi\)
−0.618487 + 0.785795i \(0.712254\pi\)
\(48\) −38.0000 − 42.4853i −0.791667 − 0.885110i
\(49\) −24.5000 + 42.4352i −0.500000 + 0.866025i
\(50\) 44.7214i 0.894427i
\(51\) 16.4758 + 78.7943i 0.323055 + 1.54499i
\(52\) −1.00000 − 1.73205i −0.0192308 − 0.0333087i
\(53\) −17.4284 + 10.0623i −0.328838 + 0.189855i −0.655325 − 0.755347i \(-0.727468\pi\)
0.326487 + 0.945202i \(0.394135\pi\)
\(54\) 60.1028 − 5.71414i 1.11302 − 0.105817i
\(55\) 25.0000 0.454545
\(56\) −40.6663 + 23.4787i −0.726184 + 0.419263i
\(57\) 32.0000 + 35.7771i 0.561404 + 0.627668i
\(58\) −17.5000 + 30.3109i −0.301724 + 0.522602i
\(59\) 17.4284 − 10.0623i 0.295397 − 0.170548i −0.344976 − 0.938611i \(-0.612113\pi\)
0.640373 + 0.768064i \(0.278780\pi\)
\(60\) 6.37298 + 2.09406i 0.106216 + 0.0349010i
\(61\) 13.0000 − 22.5167i 0.213115 − 0.369126i −0.739573 − 0.673076i \(-0.764973\pi\)
0.952688 + 0.303951i \(0.0983058\pi\)
\(62\) − 6.70820i − 0.108197i
\(63\) 7.00000 − 62.6099i 0.111111 − 0.993808i
\(64\) −41.0000 −0.640625
\(65\) −3.87298 − 2.23607i −0.0595844 − 0.0344010i
\(66\) −23.4123 + 71.2521i −0.354732 + 1.07958i
\(67\) −26.0000 − 45.0333i −0.388060 − 0.672139i 0.604129 − 0.796887i \(-0.293521\pi\)
−0.992189 + 0.124748i \(0.960188\pi\)
\(68\) −23.2379 − 13.4164i −0.341734 − 0.197300i
\(69\) 30.0000 − 26.8328i 0.434783 − 0.388881i
\(70\) 17.5000 − 30.3109i 0.250000 − 0.433013i
\(71\) − 93.9149i − 1.32274i −0.750058 − 0.661372i \(-0.769974\pi\)
0.750058 − 0.661372i \(-0.230026\pi\)
\(72\) 35.8095 − 48.6074i 0.497354 − 0.675103i
\(73\) −9.00000 − 15.5885i −0.123288 − 0.213541i 0.797775 − 0.602956i \(-0.206010\pi\)
−0.921062 + 0.389415i \(0.872677\pi\)
\(74\) 23.2379 − 13.4164i 0.314026 − 0.181303i
\(75\) −58.7298 + 12.2803i −0.783064 + 0.163738i
\(76\) −16.0000 −0.210526
\(77\) 67.7772 + 39.1312i 0.880223 + 0.508197i
\(78\) 10.0000 − 8.94427i 0.128205 − 0.114670i
\(79\) 39.5000 − 68.4160i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(80\) 36.7933 − 21.2426i 0.459917 − 0.265533i
\(81\) 24.0081 + 77.3603i 0.296396 + 0.955065i
\(82\) −35.0000 + 60.6218i −0.426829 + 0.739290i
\(83\) 140.872i 1.69726i 0.528990 + 0.848628i \(0.322571\pi\)
−0.528990 + 0.848628i \(0.677429\pi\)
\(84\) 14.0000 + 15.6525i 0.166667 + 0.186339i
\(85\) −60.0000 −0.705882
\(86\) −85.2056 − 49.1935i −0.990763 − 0.572017i
\(87\) −44.6109 − 14.6584i −0.512769 − 0.168488i
\(88\) 37.5000 + 64.9519i 0.426136 + 0.738090i
\(89\) 42.6028 + 24.5967i 0.478683 + 0.276368i 0.719868 − 0.694111i \(-0.244202\pi\)
−0.241184 + 0.970479i \(0.577536\pi\)
\(90\) −5.00000 + 44.7214i −0.0555556 + 0.496904i
\(91\) −7.00000 − 12.1244i −0.0769231 − 0.133235i
\(92\) 13.4164i 0.145831i
\(93\) 8.80948 − 1.84205i 0.0947255 − 0.0198070i
\(94\) 15.0000 + 25.9808i 0.159574 + 0.276391i
\(95\) −30.9839 + 17.8885i −0.326146 + 0.188300i
\(96\) 9.61088 + 45.9634i 0.100113 + 0.478785i
\(97\) −93.0000 −0.958763 −0.479381 − 0.877607i \(-0.659139\pi\)
−0.479381 + 0.877607i \(0.659139\pi\)
\(98\) 94.8881 − 54.7837i 0.968246 − 0.559017i
\(99\) −100.000 − 11.1803i −1.01010 − 0.112933i
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 21.3.h.b.2.1 ✓ 4
3.2 odd 2 inner 21.3.h.b.2.2 yes 4
4.3 odd 2 336.3.bn.f.65.1 4
7.2 even 3 147.3.b.d.50.2 2
7.3 odd 6 147.3.h.b.116.2 4
7.4 even 3 inner 21.3.h.b.11.2 yes 4
7.5 odd 6 147.3.b.c.50.2 2
7.6 odd 2 147.3.h.b.128.1 4
12.11 even 2 336.3.bn.f.65.2 4
21.2 odd 6 147.3.b.d.50.1 2
21.5 even 6 147.3.b.c.50.1 2
21.11 odd 6 inner 21.3.h.b.11.1 yes 4
21.17 even 6 147.3.h.b.116.1 4
21.20 even 2 147.3.h.b.128.2 4
28.11 odd 6 336.3.bn.f.305.2 4
84.11 even 6 336.3.bn.f.305.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.3.h.b.2.1 ✓ 4 1.1 even 1 trivial
21.3.h.b.2.2 yes 4 3.2 odd 2 inner
21.3.h.b.11.1 yes 4 21.11 odd 6 inner
21.3.h.b.11.2 yes 4 7.4 even 3 inner
147.3.b.c.50.1 2 21.5 even 6
147.3.b.c.50.2 2 7.5 odd 6
147.3.b.d.50.1 2 21.2 odd 6
147.3.b.d.50.2 2 7.2 even 3
147.3.h.b.116.1 4 21.17 even 6
147.3.h.b.116.2 4 7.3 odd 6
147.3.h.b.128.1 4 7.6 odd 2
147.3.h.b.128.2 4 21.20 even 2
336.3.bn.f.65.1 4 4.3 odd 2
336.3.bn.f.65.2 4 12.11 even 2
336.3.bn.f.305.1 4 84.11 even 6
336.3.bn.f.305.2 4 28.11 odd 6