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Properties

Label 63.2.h.b.25.4
Level $63$
Weight $2$
Character 63.25
Analytic conductor $0.503$
Analytic rank $0$
Dimension $10$
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] = [63,2,Mod(25,63)] 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("63.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] 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: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.991381711347.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 25.4
Root \(-0.335166 + 0.580525i\) of defining polynomial
Character \(\chi\) \(=\) 63.25
Dual form 63.2.h.b.58.4

$q$-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+0.670333 q^{2} +(1.65263 - 0.518475i) q^{3} -1.55065 q^{4} +(-0.712469 + 1.23403i) q^{5} +(1.10781 - 0.347551i) q^{6} +(-2.36039 - 1.19522i) q^{7} -2.38012 q^{8} +(2.46237 - 1.71369i) q^{9} +(-0.477591 + 0.827212i) q^{10} +(2.46539 + 4.27018i) q^{11} +(-2.56266 + 0.803975i) q^{12} +(-1.37730 - 2.38556i) q^{13} +(-1.58225 - 0.801194i) q^{14} +(-0.537632 + 2.40879i) q^{15} +1.50584 q^{16} +(0.559839 - 0.969670i) q^{17} +(1.65061 - 1.14874i) q^{18} +(-2.00752 - 3.47713i) q^{19} +(1.10479 - 1.91356i) q^{20} +(-4.52054 - 0.751449i) q^{21} +(1.65263 + 2.86244i) q^{22} +(-2.71830 + 4.70824i) q^{23} +(-3.93346 + 1.23403i) q^{24} +(1.48478 + 2.57171i) q^{25} +(-0.923251 - 1.59912i) q^{26} +(3.18087 - 4.10878i) q^{27} +(3.66015 + 1.85337i) q^{28} +(3.40555 - 5.89858i) q^{29} +(-0.360392 + 1.61469i) q^{30} +2.50584 q^{31} +5.76965 q^{32} +(6.28835 + 5.77878i) q^{33} +(0.375279 - 0.650002i) q^{34} +(3.15664 - 2.06124i) q^{35} +(-3.81828 + 2.65735i) q^{36} +(0.709787 + 1.22939i) q^{37} +(-1.34571 - 2.33083i) q^{38} +(-3.51302 - 3.22835i) q^{39} +(1.69576 - 2.93714i) q^{40} +(0.124384 + 0.215440i) q^{41} +(-3.03027 - 0.503721i) q^{42} +(-0.498313 + 0.863104i) q^{43} +(-3.82296 - 6.62156i) q^{44} +(0.360392 + 4.25959i) q^{45} +(-1.82217 + 3.15609i) q^{46} -9.47579 q^{47} +(2.48859 - 0.780738i) q^{48} +(4.14291 + 5.64237i) q^{49} +(0.995294 + 1.72390i) q^{50} +(0.422457 - 1.89277i) q^{51} +(2.13572 + 3.69917i) q^{52} +(-0.410229 + 0.710537i) q^{53} +(2.13224 - 2.75425i) q^{54} -7.02604 q^{55} +(5.61802 + 2.84476i) q^{56} +(-5.12050 - 4.70556i) q^{57} +(2.28285 - 3.95401i) q^{58} -6.58407 q^{59} +(0.833682 - 3.73521i) q^{60} +0.0752645 q^{61} +1.67974 q^{62} +(-7.86039 + 1.10192i) q^{63} +0.855913 q^{64} +3.92514 q^{65} +(4.21529 + 3.87370i) q^{66} -12.5877 q^{67} +(-0.868117 + 1.50362i) q^{68} +(-2.05125 + 9.19035i) q^{69} +(2.11600 - 1.38172i) q^{70} +0.0804951 q^{71} +(-5.86073 + 4.07880i) q^{72} +(5.34551 - 9.25869i) q^{73} +(0.475793 + 0.824098i) q^{74} +(3.78715 + 3.48026i) q^{75} +(3.11297 + 5.39183i) q^{76} +(-0.715488 - 13.0260i) q^{77} +(-2.35489 - 2.16407i) q^{78} -1.84491 q^{79} +(-1.07286 + 1.85825i) q^{80} +(3.12651 - 8.43949i) q^{81} +(0.0833788 + 0.144416i) q^{82} +(-7.23583 + 12.5328i) q^{83} +(7.00980 + 1.16524i) q^{84} +(0.797736 + 1.38172i) q^{85} +(-0.334036 + 0.578567i) q^{86} +(2.56984 - 11.5139i) q^{87} +(-5.86792 - 10.1635i) q^{88} +(6.76292 + 11.7137i) q^{89} +(0.241583 + 2.85534i) q^{90} +(0.399711 + 7.27703i) q^{91} +(4.21515 - 7.30085i) q^{92} +(4.14122 - 1.29921i) q^{93} -6.35193 q^{94} +5.72119 q^{95} +(9.53509 - 2.99142i) q^{96} +(2.70160 - 4.67930i) q^{97} +(2.77712 + 3.78226i) q^{98} +(13.3885 + 6.28982i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} - q^{3} + 8 q^{4} + 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 11 q^{9} - 7 q^{10} + 4 q^{11} - 20 q^{12} - 8 q^{13} - 20 q^{14} - 19 q^{15} - 4 q^{16} + 12 q^{17} + 4 q^{18} + q^{19} + 5 q^{20}+ \cdots + 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.670333 0.473997 0.236998 − 0.971510i \(-0.423836\pi\)
0.236998 + 0.971510i \(0.423836\pi\)
\(3\) 1.65263 − 0.518475i 0.954146 − 0.299342i
\(4\) −1.55065 −0.775327
\(5\) −0.712469 + 1.23403i −0.318626 + 0.551876i −0.980202 − 0.198002i \(-0.936555\pi\)
0.661576 + 0.749878i \(0.269888\pi\)
\(6\) 1.10781 − 0.347551i 0.452262 − 0.141887i
\(7\) −2.36039 − 1.19522i −0.892144 − 0.451750i
\(8\) −2.38012 −0.841499
\(9\) 2.46237 − 1.71369i 0.820789 − 0.571231i
\(10\) −0.477591 + 0.827212i −0.151028 + 0.261587i
\(11\) 2.46539 + 4.27018i 0.743342 + 1.28751i 0.950965 + 0.309297i \(0.100094\pi\)
−0.207623 + 0.978209i \(0.566573\pi\)
\(12\) −2.56266 + 0.803975i −0.739775 + 0.232088i
\(13\) −1.37730 − 2.38556i −0.381995 − 0.661635i 0.609352 − 0.792900i \(-0.291429\pi\)
−0.991347 + 0.131265i \(0.958096\pi\)
\(14\) −1.58225 − 0.801194i −0.422874 − 0.214128i
\(15\) −0.537632 + 2.40879i −0.138816 + 0.621948i
\(16\) 1.50584 0.376459
\(17\) 0.559839 − 0.969670i 0.135781 − 0.235180i −0.790115 − 0.612959i \(-0.789979\pi\)
0.925896 + 0.377780i \(0.123312\pi\)
\(18\) 1.65061 − 1.14874i 0.389051 − 0.270762i
\(19\) −2.00752 − 3.47713i −0.460557 − 0.797709i 0.538431 − 0.842669i \(-0.319017\pi\)
−0.998989 + 0.0449606i \(0.985684\pi\)
\(20\) 1.10479 − 1.91356i 0.247039 − 0.427884i
\(21\) −4.52054 − 0.751449i −0.986464 − 0.163980i
\(22\) 1.65263 + 2.86244i 0.352342 + 0.610274i
\(23\) −2.71830 + 4.70824i −0.566806 + 0.981736i 0.430073 + 0.902794i \(0.358488\pi\)
−0.996879 + 0.0789424i \(0.974846\pi\)
\(24\) −3.93346 + 1.23403i −0.802913 + 0.251896i
\(25\) 1.48478 + 2.57171i 0.296955 + 0.514342i
\(26\) −0.923251 − 1.59912i −0.181064 − 0.313613i
\(27\) 3.18087 − 4.10878i 0.612160 − 0.790734i
\(28\) 3.66015 + 1.85337i 0.691704 + 0.350254i
\(29\) 3.40555 − 5.89858i 0.632394 − 1.09534i −0.354667 − 0.934993i \(-0.615406\pi\)
0.987061 − 0.160346i \(-0.0512611\pi\)
\(30\) −0.360392 + 1.61469i −0.0657984 + 0.294801i
\(31\) 2.50584 0.450061 0.225031 − 0.974352i \(-0.427752\pi\)
0.225031 + 0.974352i \(0.427752\pi\)
\(32\) 5.76965 1.01994
\(33\) 6.28835 + 5.77878i 1.09466 + 1.00596i
\(34\) 0.375279 − 0.650002i 0.0643597 − 0.111474i
\(35\) 3.15664 − 2.06124i 0.533570 − 0.348414i
\(36\) −3.81828 + 2.65735i −0.636380 + 0.442891i
\(37\) 0.709787 + 1.22939i 0.116688 + 0.202110i 0.918453 − 0.395529i \(-0.129439\pi\)
−0.801765 + 0.597639i \(0.796106\pi\)
\(38\) −1.34571 − 2.33083i −0.218303 − 0.378111i
\(39\) −3.51302 − 3.22835i −0.562534 − 0.516949i
\(40\) 1.69576 − 2.93714i 0.268123 − 0.464403i
\(41\) 0.124384 + 0.215440i 0.0194256 + 0.0336460i 0.875575 − 0.483083i \(-0.160483\pi\)
−0.856149 + 0.516729i \(0.827150\pi\)
\(42\) −3.03027 − 0.503721i −0.467581 − 0.0777258i
\(43\) −0.498313 + 0.863104i −0.0759921 + 0.131622i −0.901517 − 0.432743i \(-0.857546\pi\)
0.825525 + 0.564365i \(0.190879\pi\)
\(44\) −3.82296 − 6.62156i −0.576333 − 0.998238i
\(45\) 0.360392 + 4.25959i 0.0537241 + 0.634983i
\(46\) −1.82217 + 3.15609i −0.268664 + 0.465340i
\(47\) −9.47579 −1.38219 −0.691093 − 0.722766i \(-0.742871\pi\)
−0.691093 + 0.722766i \(0.742871\pi\)
\(48\) 2.48859 − 0.780738i 0.359197 − 0.112690i
\(49\) 4.14291 + 5.64237i 0.591844 + 0.806053i
\(50\) 0.995294 + 1.72390i 0.140756 + 0.243796i
\(51\) 0.422457 − 1.89277i 0.0591559 − 0.265041i
\(52\) 2.13572 + 3.69917i 0.296171 + 0.512983i
\(53\) −0.410229 + 0.710537i −0.0563493 + 0.0975998i −0.892824 − 0.450406i \(-0.851279\pi\)
0.836475 + 0.548005i \(0.184613\pi\)
\(54\) 2.13224 − 2.75425i 0.290162 − 0.374805i
\(55\) −7.02604 −0.947392
\(56\) 5.61802 + 2.84476i 0.750739 + 0.380147i
\(57\) −5.12050 − 4.70556i −0.678226 − 0.623267i
\(58\) 2.28285 − 3.95401i 0.299753 − 0.519187i
\(59\) −6.58407 −0.857173 −0.428586 − 0.903501i \(-0.640988\pi\)
−0.428586 + 0.903501i \(0.640988\pi\)
\(60\) 0.833682 − 3.73521i 0.107628 − 0.482213i
\(61\) 0.0752645 0.00963663 0.00481831 − 0.999988i \(-0.498466\pi\)
0.00481831 + 0.999988i \(0.498466\pi\)
\(62\) 1.67974 0.213328
\(63\) −7.86039 + 1.10192i −0.990316 + 0.138829i
\(64\) 0.855913 0.106989
\(65\) 3.92514 0.486854
\(66\) 4.21529 + 3.87370i 0.518866 + 0.476820i
\(67\) −12.5877 −1.53783 −0.768916 − 0.639350i \(-0.779204\pi\)
−0.768916 + 0.639350i \(0.779204\pi\)
\(68\) −0.868117 + 1.50362i −0.105275 + 0.182341i
\(69\) −2.05125 + 9.19035i −0.246941 + 1.10639i
\(70\) 2.11600 − 1.38172i 0.252911 − 0.165147i
\(71\) 0.0804951 0.00955301 0.00477651 − 0.999989i \(-0.498480\pi\)
0.00477651 + 0.999989i \(0.498480\pi\)
\(72\) −5.86073 + 4.07880i −0.690694 + 0.480691i
\(73\) 5.34551 − 9.25869i 0.625644 − 1.08365i −0.362772 − 0.931878i \(-0.618170\pi\)
0.988416 − 0.151769i \(-0.0484971\pi\)
\(74\) 0.475793 + 0.824098i 0.0553098 + 0.0957995i
\(75\) 3.78715 + 3.48026i 0.437303 + 0.401866i
\(76\) 3.11297 + 5.39183i 0.357083 + 0.618485i
\(77\) −0.715488 − 13.0260i −0.0815374 − 1.48445i
\(78\) −2.35489 − 2.16407i −0.266639 − 0.245032i
\(79\) −1.84491 −0.207569 −0.103785 − 0.994600i \(-0.533095\pi\)
−0.103785 + 0.994600i \(0.533095\pi\)
\(80\) −1.07286 + 1.85825i −0.119950 + 0.207759i
\(81\) 3.12651 − 8.43949i 0.347390 − 0.937721i
\(82\) 0.0833788 + 0.144416i 0.00920765 + 0.0159481i
\(83\) −7.23583 + 12.5328i −0.794236 + 1.37566i 0.129088 + 0.991633i \(0.458795\pi\)
−0.923323 + 0.384023i \(0.874538\pi\)
\(84\) 7.00980 + 1.16524i 0.764832 + 0.127138i
\(85\) 0.797736 + 1.38172i 0.0865266 + 0.149868i
\(86\) −0.334036 + 0.578567i −0.0360200 + 0.0623885i
\(87\) 2.56984 − 11.5139i 0.275516 − 1.23442i
\(88\) −5.86792 − 10.1635i −0.625522 − 1.08344i
\(89\) 6.76292 + 11.7137i 0.716868 + 1.24165i 0.962235 + 0.272222i \(0.0877584\pi\)
−0.245366 + 0.969430i \(0.578908\pi\)
\(90\) 0.241583 + 2.85534i 0.0254651 + 0.300980i
\(91\) 0.399711 + 7.27703i 0.0419011 + 0.762840i
\(92\) 4.21515 − 7.30085i 0.439460 − 0.761167i
\(93\) 4.14122 − 1.29921i 0.429424 − 0.134722i
\(94\) −6.35193 −0.655152
\(95\) 5.72119 0.586982
\(96\) 9.53509 − 2.99142i 0.973171 − 0.305310i
\(97\) 2.70160 − 4.67930i 0.274306 − 0.475111i −0.695654 − 0.718377i \(-0.744885\pi\)
0.969960 + 0.243266i \(0.0782187\pi\)
\(98\) 2.77712 + 3.78226i 0.280532 + 0.382066i
\(99\) 13.3885 + 6.28982i 1.34559 + 0.632151i
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 63.2.h.b.25.4 yes 10
3.2 odd 2 189.2.h.b.46.2 10
4.3 odd 2 1008.2.q.i.529.1 10
7.2 even 3 63.2.g.b.16.2 yes 10
7.3 odd 6 441.2.f.f.295.2 10
7.4 even 3 441.2.f.e.295.2 10
7.5 odd 6 441.2.g.f.79.2 10
7.6 odd 2 441.2.h.f.214.4 10
9.2 odd 6 567.2.e.e.487.4 10
9.4 even 3 63.2.g.b.4.2 ✓ 10
9.5 odd 6 189.2.g.b.172.4 10
9.7 even 3 567.2.e.f.487.2 10
12.11 even 2 3024.2.q.i.2881.5 10
21.2 odd 6 189.2.g.b.100.4 10
21.5 even 6 1323.2.g.f.667.4 10
21.11 odd 6 1323.2.f.e.883.4 10
21.17 even 6 1323.2.f.f.883.4 10
21.20 even 2 1323.2.h.f.802.2 10
28.23 odd 6 1008.2.t.i.961.4 10
36.23 even 6 3024.2.t.i.1873.1 10
36.31 odd 6 1008.2.t.i.193.4 10
63.2 odd 6 567.2.e.e.163.4 10
63.4 even 3 441.2.f.e.148.2 10
63.5 even 6 1323.2.h.f.226.2 10
63.11 odd 6 3969.2.a.bc.1.2 5
63.13 odd 6 441.2.g.f.67.2 10
63.16 even 3 567.2.e.f.163.2 10
63.23 odd 6 189.2.h.b.37.2 10
63.25 even 3 3969.2.a.z.1.4 5
63.31 odd 6 441.2.f.f.148.2 10
63.32 odd 6 1323.2.f.e.442.4 10
63.38 even 6 3969.2.a.bb.1.2 5
63.40 odd 6 441.2.h.f.373.4 10
63.41 even 6 1323.2.g.f.361.4 10
63.52 odd 6 3969.2.a.ba.1.4 5
63.58 even 3 inner 63.2.h.b.58.4 yes 10
63.59 even 6 1323.2.f.f.442.4 10
84.23 even 6 3024.2.t.i.289.1 10
252.23 even 6 3024.2.q.i.2305.5 10
252.247 odd 6 1008.2.q.i.625.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.g.b.4.2 ✓ 10 9.4 even 3
63.2.g.b.16.2 yes 10 7.2 even 3
63.2.h.b.25.4 yes 10 1.1 even 1 trivial
63.2.h.b.58.4 yes 10 63.58 even 3 inner
189.2.g.b.100.4 10 21.2 odd 6
189.2.g.b.172.4 10 9.5 odd 6
189.2.h.b.37.2 10 63.23 odd 6
189.2.h.b.46.2 10 3.2 odd 2
441.2.f.e.148.2 10 63.4 even 3
441.2.f.e.295.2 10 7.4 even 3
441.2.f.f.148.2 10 63.31 odd 6
441.2.f.f.295.2 10 7.3 odd 6
441.2.g.f.67.2 10 63.13 odd 6
441.2.g.f.79.2 10 7.5 odd 6
441.2.h.f.214.4 10 7.6 odd 2
441.2.h.f.373.4 10 63.40 odd 6
567.2.e.e.163.4 10 63.2 odd 6
567.2.e.e.487.4 10 9.2 odd 6
567.2.e.f.163.2 10 63.16 even 3
567.2.e.f.487.2 10 9.7 even 3
1008.2.q.i.529.1 10 4.3 odd 2
1008.2.q.i.625.1 10 252.247 odd 6
1008.2.t.i.193.4 10 36.31 odd 6
1008.2.t.i.961.4 10 28.23 odd 6
1323.2.f.e.442.4 10 63.32 odd 6
1323.2.f.e.883.4 10 21.11 odd 6
1323.2.f.f.442.4 10 63.59 even 6
1323.2.f.f.883.4 10 21.17 even 6
1323.2.g.f.361.4 10 63.41 even 6
1323.2.g.f.667.4 10 21.5 even 6
1323.2.h.f.226.2 10 63.5 even 6
1323.2.h.f.802.2 10 21.20 even 2
3024.2.q.i.2305.5 10 252.23 even 6
3024.2.q.i.2881.5 10 12.11 even 2
3024.2.t.i.289.1 10 84.23 even 6
3024.2.t.i.1873.1 10 36.23 even 6
3969.2.a.z.1.4 5 63.25 even 3
3969.2.a.ba.1.4 5 63.52 odd 6
3969.2.a.bb.1.2 5 63.38 even 6
3969.2.a.bc.1.2 5 63.11 odd 6