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Properties

Label 63.4.e.a.37.1
Level $63$
Weight $4$
Character 63.37
Analytic conductor $3.717$
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] = [63,4,Mod(37,63)] 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("63.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71712033036\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 63.37
Dual form 63.4.e.a.46.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 2.59808i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(-3.50000 + 18.1865i) q^{7} -21.0000 q^{8} +(-4.50000 - 7.79423i) q^{10} +(-7.50000 - 12.9904i) q^{11} -64.0000 q^{13} +(-42.0000 - 36.3731i) q^{14} +(35.5000 - 61.4878i) q^{16} +(42.0000 + 72.7461i) q^{17} +(8.00000 - 13.8564i) q^{19} +3.00000 q^{20} +45.0000 q^{22} +(-42.0000 + 72.7461i) q^{23} +(58.0000 + 100.459i) q^{25} +(96.0000 - 166.277i) q^{26} +(17.5000 - 6.06218i) q^{28} +297.000 q^{29} +(126.500 + 219.104i) q^{31} +(22.5000 + 38.9711i) q^{32} -252.000 q^{34} +(-42.0000 - 36.3731i) q^{35} +(158.000 - 273.664i) q^{37} +(24.0000 + 41.5692i) q^{38} +(31.5000 - 54.5596i) q^{40} -360.000 q^{41} +26.0000 q^{43} +(-7.50000 + 12.9904i) q^{44} +(-126.000 - 218.238i) q^{46} +(-15.0000 + 25.9808i) q^{47} +(-318.500 - 127.306i) q^{49} -348.000 q^{50} +(32.0000 + 55.4256i) q^{52} +(181.500 + 314.367i) q^{53} +45.0000 q^{55} +(73.5000 - 381.917i) q^{56} +(-445.500 + 771.629i) q^{58} +(-7.50000 - 12.9904i) q^{59} +(59.0000 - 102.191i) q^{61} -759.000 q^{62} +433.000 q^{64} +(96.0000 - 166.277i) q^{65} +(185.000 + 320.429i) q^{67} +(42.0000 - 72.7461i) q^{68} +(157.500 - 54.5596i) q^{70} +342.000 q^{71} +(-181.000 - 313.501i) q^{73} +(474.000 + 820.992i) q^{74} -16.0000 q^{76} +(262.500 - 90.9327i) q^{77} +(-233.500 + 404.434i) q^{79} +(106.500 + 184.463i) q^{80} +(540.000 - 935.307i) q^{82} -477.000 q^{83} -252.000 q^{85} +(-39.0000 + 67.5500i) q^{86} +(157.500 + 272.798i) q^{88} +(453.000 - 784.619i) q^{89} +(224.000 - 1163.94i) q^{91} +84.0000 q^{92} +(-45.0000 - 77.9423i) q^{94} +(24.0000 + 41.5692i) q^{95} +503.000 q^{97} +(808.500 - 636.529i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - q^{4} - 3 q^{5} - 7 q^{7} - 42 q^{8} - 9 q^{10} - 15 q^{11} - 128 q^{13} - 84 q^{14} + 71 q^{16} + 84 q^{17} + 16 q^{19} + 6 q^{20} + 90 q^{22} - 84 q^{23} + 116 q^{25} + 192 q^{26} + 35 q^{28}+ \cdots + 1617 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.50000 + 2.59808i −0.530330 + 0.918559i 0.469044 + 0.883175i \(0.344599\pi\)
−0.999374 + 0.0353837i \(0.988735\pi\)
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(5\) −1.50000 + 2.59808i −0.134164 + 0.232379i −0.925278 − 0.379290i \(-0.876168\pi\)
0.791114 + 0.611669i \(0.209502\pi\)
\(6\) 0 0
\(7\) −3.50000 + 18.1865i −0.188982 + 0.981981i
\(8\) −21.0000 −0.928078
\(9\) 0 0
\(10\) −4.50000 − 7.79423i −0.142302 − 0.246475i
\(11\) −7.50000 − 12.9904i −0.205576 − 0.356068i 0.744740 − 0.667355i \(-0.232573\pi\)
−0.950316 + 0.311287i \(0.899240\pi\)
\(12\) 0 0
\(13\) −64.0000 −1.36542 −0.682708 − 0.730691i \(-0.739198\pi\)
−0.682708 + 0.730691i \(0.739198\pi\)
\(14\) −42.0000 − 36.3731i −0.801784 − 0.694365i
\(15\) 0 0
\(16\) 35.5000 − 61.4878i 0.554688 − 0.960747i
\(17\) 42.0000 + 72.7461i 0.599206 + 1.03785i 0.992939 + 0.118630i \(0.0378502\pi\)
−0.393733 + 0.919225i \(0.628817\pi\)
\(18\) 0 0
\(19\) 8.00000 − 13.8564i 0.0965961 − 0.167309i −0.813678 − 0.581317i \(-0.802538\pi\)
0.910274 + 0.414007i \(0.135871\pi\)
\(20\) 3.00000 0.0335410
\(21\) 0 0
\(22\) 45.0000 0.436092
\(23\) −42.0000 + 72.7461i −0.380765 + 0.659505i −0.991172 − 0.132583i \(-0.957673\pi\)
0.610406 + 0.792088i \(0.291006\pi\)
\(24\) 0 0
\(25\) 58.0000 + 100.459i 0.464000 + 0.803672i
\(26\) 96.0000 − 166.277i 0.724121 − 1.25421i
\(27\) 0 0
\(28\) 17.5000 − 6.06218i 0.118114 − 0.0409159i
\(29\) 297.000 1.90178 0.950888 − 0.309535i \(-0.100173\pi\)
0.950888 + 0.309535i \(0.100173\pi\)
\(30\) 0 0
\(31\) 126.500 + 219.104i 0.732906 + 1.26943i 0.955636 + 0.294550i \(0.0951696\pi\)
−0.222731 + 0.974880i \(0.571497\pi\)
\(32\) 22.5000 + 38.9711i 0.124296 + 0.215287i
\(33\) 0 0
\(34\) −252.000 −1.27111
\(35\) −42.0000 − 36.3731i −0.202837 − 0.175662i
\(36\) 0 0
\(37\) 158.000 − 273.664i 0.702028 − 1.21595i −0.265725 − 0.964049i \(-0.585611\pi\)
0.967753 − 0.251900i \(-0.0810553\pi\)
\(38\) 24.0000 + 41.5692i 0.102456 + 0.177458i
\(39\) 0 0
\(40\) 31.5000 − 54.5596i 0.124515 − 0.215666i
\(41\) −360.000 −1.37128 −0.685641 − 0.727940i \(-0.740478\pi\)
−0.685641 + 0.727940i \(0.740478\pi\)
\(42\) 0 0
\(43\) 26.0000 0.0922084 0.0461042 − 0.998937i \(-0.485319\pi\)
0.0461042 + 0.998937i \(0.485319\pi\)
\(44\) −7.50000 + 12.9904i −0.0256970 + 0.0445085i
\(45\) 0 0
\(46\) −126.000 − 218.238i −0.403863 − 0.699511i
\(47\) −15.0000 + 25.9808i −0.0465527 + 0.0806316i −0.888363 − 0.459142i \(-0.848157\pi\)
0.841810 + 0.539774i \(0.181490\pi\)
\(48\) 0 0
\(49\) −318.500 − 127.306i −0.928571 − 0.371154i
\(50\) −348.000 −0.984293
\(51\) 0 0
\(52\) 32.0000 + 55.4256i 0.0853385 + 0.147811i
\(53\) 181.500 + 314.367i 0.470395 + 0.814748i 0.999427 − 0.0338538i \(-0.0107781\pi\)
−0.529032 + 0.848602i \(0.677445\pi\)
\(54\) 0 0
\(55\) 45.0000 0.110324
\(56\) 73.5000 − 381.917i 0.175390 − 0.911354i
\(57\) 0 0
\(58\) −445.500 + 771.629i −1.00857 + 1.74689i
\(59\) −7.50000 − 12.9904i −0.0165494 − 0.0286645i 0.857632 − 0.514264i \(-0.171935\pi\)
−0.874182 + 0.485599i \(0.838601\pi\)
\(60\) 0 0
\(61\) 59.0000 − 102.191i 0.123839 − 0.214495i −0.797440 − 0.603399i \(-0.793813\pi\)
0.921279 + 0.388903i \(0.127146\pi\)
\(62\) −759.000 −1.55473
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) 96.0000 − 166.277i 0.183190 − 0.317294i
\(66\) 0 0
\(67\) 185.000 + 320.429i 0.337334 + 0.584279i 0.983930 − 0.178553i \(-0.0571417\pi\)
−0.646597 + 0.762832i \(0.723808\pi\)
\(68\) 42.0000 − 72.7461i 0.0749007 − 0.129732i
\(69\) 0 0
\(70\) 157.500 − 54.5596i 0.268926 − 0.0931589i
\(71\) 342.000 0.571661 0.285831 − 0.958280i \(-0.407731\pi\)
0.285831 + 0.958280i \(0.407731\pi\)
\(72\) 0 0
\(73\) −181.000 − 313.501i −0.290198 − 0.502638i 0.683658 − 0.729802i \(-0.260388\pi\)
−0.973856 + 0.227165i \(0.927054\pi\)
\(74\) 474.000 + 820.992i 0.744613 + 1.28971i
\(75\) 0 0
\(76\) −16.0000 −0.0241490
\(77\) 262.500 − 90.9327i 0.388502 − 0.134581i
\(78\) 0 0
\(79\) −233.500 + 404.434i −0.332542 + 0.575979i −0.983010 − 0.183555i \(-0.941240\pi\)
0.650468 + 0.759534i \(0.274573\pi\)
\(80\) 106.500 + 184.463i 0.148838 + 0.257795i
\(81\) 0 0
\(82\) 540.000 − 935.307i 0.727232 − 1.25960i
\(83\) −477.000 −0.630814 −0.315407 − 0.948957i \(-0.602141\pi\)
−0.315407 + 0.948957i \(0.602141\pi\)
\(84\) 0 0
\(85\) −252.000 −0.321568
\(86\) −39.0000 + 67.5500i −0.0489009 + 0.0846989i
\(87\) 0 0
\(88\) 157.500 + 272.798i 0.190790 + 0.330459i
\(89\) 453.000 − 784.619i 0.539527 − 0.934488i −0.459402 − 0.888228i \(-0.651936\pi\)
0.998929 − 0.0462600i \(-0.0147303\pi\)
\(90\) 0 0
\(91\) 224.000 − 1163.94i 0.258039 − 1.34081i
\(92\) 84.0000 0.0951914
\(93\) 0 0
\(94\) −45.0000 − 77.9423i −0.0493765 − 0.0855227i
\(95\) 24.0000 + 41.5692i 0.0259195 + 0.0448938i
\(96\) 0 0
\(97\) 503.000 0.526515 0.263257 − 0.964726i \(-0.415203\pi\)
0.263257 + 0.964726i \(0.415203\pi\)
\(98\) 808.500 − 636.529i 0.833376 − 0.656113i
\(99\) 0 0
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 63.4.e.a.37.1 2
3.2 odd 2 21.4.e.a.16.1 yes 2
7.2 even 3 441.4.a.l.1.1 1
7.3 odd 6 441.4.e.c.361.1 2
7.4 even 3 inner 63.4.e.a.46.1 2
7.5 odd 6 441.4.a.k.1.1 1
7.6 odd 2 441.4.e.c.226.1 2
12.11 even 2 336.4.q.e.289.1 2
21.2 odd 6 147.4.a.b.1.1 1
21.5 even 6 147.4.a.a.1.1 1
21.11 odd 6 21.4.e.a.4.1 ✓ 2
21.17 even 6 147.4.e.h.67.1 2
21.20 even 2 147.4.e.h.79.1 2
84.11 even 6 336.4.q.e.193.1 2
84.23 even 6 2352.4.a.i.1.1 1
84.47 odd 6 2352.4.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.a.4.1 ✓ 2 21.11 odd 6
21.4.e.a.16.1 yes 2 3.2 odd 2
63.4.e.a.37.1 2 1.1 even 1 trivial
63.4.e.a.46.1 2 7.4 even 3 inner
147.4.a.a.1.1 1 21.5 even 6
147.4.a.b.1.1 1 21.2 odd 6
147.4.e.h.67.1 2 21.17 even 6
147.4.e.h.79.1 2 21.20 even 2
336.4.q.e.193.1 2 84.11 even 6
336.4.q.e.289.1 2 12.11 even 2
441.4.a.k.1.1 1 7.5 odd 6
441.4.a.l.1.1 1 7.2 even 3
441.4.e.c.226.1 2 7.6 odd 2
441.4.e.c.361.1 2 7.3 odd 6
2352.4.a.i.1.1 1 84.23 even 6
2352.4.a.bd.1.1 1 84.47 odd 6