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Properties

Label 21.3.d.a.13.2
Level $21$
Weight $3$
Character 21.13
Analytic conductor $0.572$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(13,21)] 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("21.13"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 21.d (of order \(2\), degree \(1\), 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: \(0.572208555157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 13.2
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.13
Dual form 21.3.d.a.13.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.73205i q^{3} -3.00000 q^{4} -6.92820i q^{5} +1.73205i q^{6} +(1.00000 + 6.92820i) q^{7} -7.00000 q^{8} -3.00000 q^{9} -6.92820i q^{10} +10.0000 q^{11} -5.19615i q^{12} +6.92820i q^{13} +(1.00000 + 6.92820i) q^{14} +12.0000 q^{15} +5.00000 q^{16} -3.00000 q^{18} -20.7846i q^{19} +20.7846i q^{20} +(-12.0000 + 1.73205i) q^{21} +10.0000 q^{22} -14.0000 q^{23} -12.1244i q^{24} -23.0000 q^{25} +6.92820i q^{26} -5.19615i q^{27} +(-3.00000 - 20.7846i) q^{28} -38.0000 q^{29} +12.0000 q^{30} +27.7128i q^{31} +33.0000 q^{32} +17.3205i q^{33} +(48.0000 - 6.92820i) q^{35} +9.00000 q^{36} +26.0000 q^{37} -20.7846i q^{38} -12.0000 q^{39} +48.4974i q^{40} -69.2820i q^{41} +(-12.0000 + 1.73205i) q^{42} +26.0000 q^{43} -30.0000 q^{44} +20.7846i q^{45} -14.0000 q^{46} +27.7128i q^{47} +8.66025i q^{48} +(-47.0000 + 13.8564i) q^{49} -23.0000 q^{50} -20.7846i q^{52} +10.0000 q^{53} -5.19615i q^{54} -69.2820i q^{55} +(-7.00000 - 48.4974i) q^{56} +36.0000 q^{57} -38.0000 q^{58} +76.2102i q^{59} -36.0000 q^{60} +34.6410i q^{61} +27.7128i q^{62} +(-3.00000 - 20.7846i) q^{63} +13.0000 q^{64} +48.0000 q^{65} +17.3205i q^{66} +74.0000 q^{67} -24.2487i q^{69} +(48.0000 - 6.92820i) q^{70} -62.0000 q^{71} +21.0000 q^{72} -41.5692i q^{73} +26.0000 q^{74} -39.8372i q^{75} +62.3538i q^{76} +(10.0000 + 69.2820i) q^{77} -12.0000 q^{78} -46.0000 q^{79} -34.6410i q^{80} +9.00000 q^{81} -69.2820i q^{82} +90.0666i q^{83} +(36.0000 - 5.19615i) q^{84} +26.0000 q^{86} -65.8179i q^{87} -70.0000 q^{88} -41.5692i q^{89} +20.7846i q^{90} +(-48.0000 + 6.92820i) q^{91} +42.0000 q^{92} -48.0000 q^{93} +27.7128i q^{94} -144.000 q^{95} +57.1577i q^{96} +55.4256i q^{97} +(-47.0000 + 13.8564i) q^{98} -30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 6 q^{4} + 2 q^{7} - 14 q^{8} - 6 q^{9} + 20 q^{11} + 2 q^{14} + 24 q^{15} + 10 q^{16} - 6 q^{18} - 24 q^{21} + 20 q^{22} - 28 q^{23} - 46 q^{25} - 6 q^{28} - 76 q^{29} + 24 q^{30} + 66 q^{32}+ \cdots - 60 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.500000 0.250000 − 0.968246i \(-0.419569\pi\)
0.250000 + 0.968246i \(0.419569\pi\)
\(3\) 1.73205i 0.577350i
\(4\) −3.00000 −0.750000
\(5\) − 6.92820i − 1.38564i −0.721110 − 0.692820i \(-0.756368\pi\)
0.721110 − 0.692820i \(-0.243632\pi\)
\(6\) 1.73205i 0.288675i
\(7\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(8\) −7.00000 −0.875000
\(9\) −3.00000 −0.333333
\(10\) − 6.92820i − 0.692820i
\(11\) 10.0000 0.909091 0.454545 − 0.890724i \(-0.349802\pi\)
0.454545 + 0.890724i \(0.349802\pi\)
\(12\) − 5.19615i − 0.433013i
\(13\) 6.92820i 0.532939i 0.963843 + 0.266469i \(0.0858571\pi\)
−0.963843 + 0.266469i \(0.914143\pi\)
\(14\) 1.00000 + 6.92820i 0.0714286 + 0.494872i
\(15\) 12.0000 0.800000
\(16\) 5.00000 0.312500
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −3.00000 −0.166667
\(19\) − 20.7846i − 1.09393i −0.837157 − 0.546963i \(-0.815784\pi\)
0.837157 − 0.546963i \(-0.184216\pi\)
\(20\) 20.7846i 1.03923i
\(21\) −12.0000 + 1.73205i −0.571429 + 0.0824786i
\(22\) 10.0000 0.454545
\(23\) −14.0000 −0.608696 −0.304348 − 0.952561i \(-0.598439\pi\)
−0.304348 + 0.952561i \(0.598439\pi\)
\(24\) − 12.1244i − 0.505181i
\(25\) −23.0000 −0.920000
\(26\) 6.92820i 0.266469i
\(27\) − 5.19615i − 0.192450i
\(28\) −3.00000 − 20.7846i −0.107143 − 0.742307i
\(29\) −38.0000 −1.31034 −0.655172 − 0.755479i \(-0.727404\pi\)
−0.655172 + 0.755479i \(0.727404\pi\)
\(30\) 12.0000 0.400000
\(31\) 27.7128i 0.893962i 0.894544 + 0.446981i \(0.147501\pi\)
−0.894544 + 0.446981i \(0.852499\pi\)
\(32\) 33.0000 1.03125
\(33\) 17.3205i 0.524864i
\(34\) 0 0
\(35\) 48.0000 − 6.92820i 1.37143 − 0.197949i
\(36\) 9.00000 0.250000
\(37\) 26.0000 0.702703 0.351351 − 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) − 20.7846i − 0.546963i
\(39\) −12.0000 −0.307692
\(40\) 48.4974i 1.21244i
\(41\) − 69.2820i − 1.68981i −0.534920 − 0.844903i \(-0.679658\pi\)
0.534920 − 0.844903i \(-0.320342\pi\)
\(42\) −12.0000 + 1.73205i −0.285714 + 0.0412393i
\(43\) 26.0000 0.604651 0.302326 − 0.953205i \(-0.402237\pi\)
0.302326 + 0.953205i \(0.402237\pi\)
\(44\) −30.0000 −0.681818
\(45\) 20.7846i 0.461880i
\(46\) −14.0000 −0.304348
\(47\) 27.7128i 0.589634i 0.955554 + 0.294817i \(0.0952587\pi\)
−0.955554 + 0.294817i \(0.904741\pi\)
\(48\) 8.66025i 0.180422i
\(49\) −47.0000 + 13.8564i −0.959184 + 0.282784i
\(50\) −23.0000 −0.460000
\(51\) 0 0
\(52\) − 20.7846i − 0.399704i
\(53\) 10.0000 0.188679 0.0943396 − 0.995540i \(-0.469926\pi\)
0.0943396 + 0.995540i \(0.469926\pi\)
\(54\) − 5.19615i − 0.0962250i
\(55\) − 69.2820i − 1.25967i
\(56\) −7.00000 − 48.4974i −0.125000 − 0.866025i
\(57\) 36.0000 0.631579
\(58\) −38.0000 −0.655172
\(59\) 76.2102i 1.29170i 0.763465 + 0.645849i \(0.223497\pi\)
−0.763465 + 0.645849i \(0.776503\pi\)
\(60\) −36.0000 −0.600000
\(61\) 34.6410i 0.567886i 0.958841 + 0.283943i \(0.0916426\pi\)
−0.958841 + 0.283943i \(0.908357\pi\)
\(62\) 27.7128i 0.446981i
\(63\) −3.00000 − 20.7846i −0.0476190 − 0.329914i
\(64\) 13.0000 0.203125
\(65\) 48.0000 0.738462
\(66\) 17.3205i 0.262432i
\(67\) 74.0000 1.10448 0.552239 − 0.833686i \(-0.313774\pi\)
0.552239 + 0.833686i \(0.313774\pi\)
\(68\) 0 0
\(69\) − 24.2487i − 0.351431i
\(70\) 48.0000 − 6.92820i 0.685714 − 0.0989743i
\(71\) −62.0000 −0.873239 −0.436620 − 0.899646i \(-0.643824\pi\)
−0.436620 + 0.899646i \(0.643824\pi\)
\(72\) 21.0000 0.291667
\(73\) − 41.5692i − 0.569441i −0.958611 − 0.284721i \(-0.908099\pi\)
0.958611 − 0.284721i \(-0.0919009\pi\)
\(74\) 26.0000 0.351351
\(75\) − 39.8372i − 0.531162i
\(76\) 62.3538i 0.820445i
\(77\) 10.0000 + 69.2820i 0.129870 + 0.899767i
\(78\) −12.0000 −0.153846
\(79\) −46.0000 −0.582278 −0.291139 − 0.956681i \(-0.594034\pi\)
−0.291139 + 0.956681i \(0.594034\pi\)
\(80\) − 34.6410i − 0.433013i
\(81\) 9.00000 0.111111
\(82\) − 69.2820i − 0.844903i
\(83\) 90.0666i 1.08514i 0.840011 + 0.542570i \(0.182549\pi\)
−0.840011 + 0.542570i \(0.817451\pi\)
\(84\) 36.0000 − 5.19615i 0.428571 − 0.0618590i
\(85\) 0 0
\(86\) 26.0000 0.302326
\(87\) − 65.8179i − 0.756528i
\(88\) −70.0000 −0.795455
\(89\) − 41.5692i − 0.467070i −0.972348 − 0.233535i \(-0.924971\pi\)
0.972348 − 0.233535i \(-0.0750293\pi\)
\(90\) 20.7846i 0.230940i
\(91\) −48.0000 + 6.92820i −0.527473 + 0.0761341i
\(92\) 42.0000 0.456522
\(93\) −48.0000 −0.516129
\(94\) 27.7128i 0.294817i
\(95\) −144.000 −1.51579
\(96\) 57.1577i 0.595392i
\(97\) 55.4256i 0.571398i 0.958319 + 0.285699i \(0.0922258\pi\)
−0.958319 + 0.285699i \(0.907774\pi\)
\(98\) −47.0000 + 13.8564i −0.479592 + 0.141392i
\(99\) −30.0000 −0.303030
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.d.a.13.2 yes 2
3.2 odd 2 63.3.d.b.55.2 2
4.3 odd 2 336.3.f.a.97.1 2
5.2 odd 4 525.3.e.a.349.4 4
5.3 odd 4 525.3.e.a.349.1 4
5.4 even 2 525.3.h.a.76.1 2
7.2 even 3 147.3.f.b.31.1 2
7.3 odd 6 147.3.f.b.19.1 2
7.4 even 3 147.3.f.d.19.1 2
7.5 odd 6 147.3.f.d.31.1 2
7.6 odd 2 inner 21.3.d.a.13.1 ✓ 2
8.3 odd 2 1344.3.f.b.769.2 2
8.5 even 2 1344.3.f.c.769.1 2
12.11 even 2 1008.3.f.d.433.2 2
21.2 odd 6 441.3.m.f.325.1 2
21.5 even 6 441.3.m.d.325.1 2
21.11 odd 6 441.3.m.d.19.1 2
21.17 even 6 441.3.m.f.19.1 2
21.20 even 2 63.3.d.b.55.1 2
28.27 even 2 336.3.f.a.97.2 2
35.13 even 4 525.3.e.a.349.2 4
35.27 even 4 525.3.e.a.349.3 4
35.34 odd 2 525.3.h.a.76.2 2
56.13 odd 2 1344.3.f.c.769.2 2
56.27 even 2 1344.3.f.b.769.1 2
84.83 odd 2 1008.3.f.d.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.3.d.a.13.1 ✓ 2 7.6 odd 2 inner
21.3.d.a.13.2 yes 2 1.1 even 1 trivial
63.3.d.b.55.1 2 21.20 even 2
63.3.d.b.55.2 2 3.2 odd 2
147.3.f.b.19.1 2 7.3 odd 6
147.3.f.b.31.1 2 7.2 even 3
147.3.f.d.19.1 2 7.4 even 3
147.3.f.d.31.1 2 7.5 odd 6
336.3.f.a.97.1 2 4.3 odd 2
336.3.f.a.97.2 2 28.27 even 2
441.3.m.d.19.1 2 21.11 odd 6
441.3.m.d.325.1 2 21.5 even 6
441.3.m.f.19.1 2 21.17 even 6
441.3.m.f.325.1 2 21.2 odd 6
525.3.e.a.349.1 4 5.3 odd 4
525.3.e.a.349.2 4 35.13 even 4
525.3.e.a.349.3 4 35.27 even 4
525.3.e.a.349.4 4 5.2 odd 4
525.3.h.a.76.1 2 5.4 even 2
525.3.h.a.76.2 2 35.34 odd 2
1008.3.f.d.433.1 2 84.83 odd 2
1008.3.f.d.433.2 2 12.11 even 2
1344.3.f.b.769.1 2 56.27 even 2
1344.3.f.b.769.2 2 8.3 odd 2
1344.3.f.c.769.1 2 8.5 even 2
1344.3.f.c.769.2 2 56.13 odd 2