Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,7,Mod(10,63)] 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("63.10"); S:= CuspForms(chi, 7); 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, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 63.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-12,0,-80,-315] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.4934072681\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 10.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 63.10
Dual form 63.7.m.a.19.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-6.00000 - 10.3923i) q^{2} +(-40.0000 + 69.2820i) q^{4} +(-157.500 + 90.9327i) q^{5} -343.000 q^{7} +192.000 q^{8} +(1890.00 + 1091.19i) q^{10} +(739.500 - 1280.85i) q^{11} +484.974i q^{13} +(2058.00 + 3564.56i) q^{14} +(1408.00 + 2438.73i) q^{16} +(2614.50 + 1509.48i) q^{17} +(5953.50 - 3437.25i) q^{19} -14549.2i q^{20} -17748.0 q^{22} +(-2956.50 - 5120.81i) q^{23} +(8725.00 - 15112.1i) q^{25} +(5040.00 - 2909.85i) q^{26} +(13720.0 - 23763.7i) q^{28} -3978.00 q^{29} +(-11098.5 - 6407.72i) q^{31} +(23040.0 - 39906.5i) q^{32} -36227.6i q^{34} +(54022.5 - 31189.9i) q^{35} +(30788.5 + 53327.2i) q^{37} +(-71442.0 - 41247.1i) q^{38} +(-30240.0 + 17459.1i) q^{40} +110574. i q^{41} -17414.0 q^{43} +(59160.0 + 102468. i) q^{44} +(-35478.0 + 61449.7i) q^{46} +(26554.5 - 15331.2i) q^{47} +117649. q^{49} -209400. q^{50} +(-33600.0 - 19399.0i) q^{52} +(-30256.5 + 52405.8i) q^{53} +268979. i q^{55} -65856.0 q^{56} +(23868.0 + 41340.6i) q^{58} +(186826. + 107864. i) q^{59} +(140942. - 81372.6i) q^{61} +153785. i q^{62} -372736. q^{64} +(-44100.0 - 76383.4i) q^{65} +(134388. - 232768. i) q^{67} +(-209160. + 120759. i) q^{68} +(-648270. - 374279. i) q^{70} -101922. q^{71} +(275090. + 158823. i) q^{73} +(369462. - 639927. i) q^{74} +549961. i q^{76} +(-253648. + 439332. i) q^{77} +(-181116. - 313701. i) q^{79} +(-443520. - 256066. i) q^{80} +(1.14912e6 - 663445. i) q^{82} -216783. i q^{83} -549045. q^{85} +(104484. + 180972. i) q^{86} +(141984. - 245924. i) q^{88} +(1.15577e6 - 667282. i) q^{89} -166346. i q^{91} +473040. q^{92} +(-318654. - 183975. i) q^{94} +(-625118. + 1.08274e6i) q^{95} +1.51409e6i q^{97} +(-705894. - 1.22264e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{2} - 80 q^{4} - 315 q^{5} - 686 q^{7} + 384 q^{8} + 3780 q^{10} + 1479 q^{11} + 4116 q^{14} + 2816 q^{16} + 5229 q^{17} + 11907 q^{19} - 35496 q^{22} - 5913 q^{23} + 17450 q^{25} + 10080 q^{26}+ \cdots - 1411788 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.00000 − 10.3923i −0.750000 − 1.29904i −0.947822 − 0.318800i \(-0.896720\pi\)
0.197822 − 0.980238i \(-0.436613\pi\)
\(3\) 0 0
\(4\) −40.0000 + 69.2820i −0.625000 + 1.08253i
\(5\) −157.500 + 90.9327i −1.26000 + 0.727461i −0.973074 − 0.230492i \(-0.925967\pi\)
−0.286926 + 0.957953i \(0.592633\pi\)
\(6\) 0 0
\(7\) −343.000 −1.00000
\(8\) 192.000 0.375000
\(9\) 0 0
\(10\) 1890.00 + 1091.19i 1.89000 + 1.09119i
\(11\) 739.500 − 1280.85i 0.555597 − 0.962323i −0.442259 − 0.896887i \(-0.645823\pi\)
0.997857 − 0.0654356i \(-0.0208437\pi\)
\(12\) 0 0
\(13\) 484.974i 0.220744i 0.993890 + 0.110372i \(0.0352042\pi\)
−0.993890 + 0.110372i \(0.964796\pi\)
\(14\) 2058.00 + 3564.56i 0.750000 + 1.29904i
\(15\) 0 0
\(16\) 1408.00 + 2438.73i 0.343750 + 0.595392i
\(17\) 2614.50 + 1509.48i 0.532160 + 0.307242i 0.741895 − 0.670516i \(-0.233927\pi\)
−0.209736 + 0.977758i \(0.567260\pi\)
\(18\) 0 0
\(19\) 5953.50 − 3437.25i 0.867984 − 0.501131i 0.00130600 − 0.999999i \(-0.499584\pi\)
0.866678 + 0.498869i \(0.166251\pi\)
\(20\) − 14549.2i − 1.81865i
\(21\) 0 0
\(22\) −17748.0 −1.66679
\(23\) −2956.50 − 5120.81i −0.242993 − 0.420877i 0.718572 − 0.695452i \(-0.244796\pi\)
−0.961566 + 0.274576i \(0.911463\pi\)
\(24\) 0 0
\(25\) 8725.00 − 15112.1i 0.558400 − 0.967177i
\(26\) 5040.00 − 2909.85i 0.286755 − 0.165558i
\(27\) 0 0
\(28\) 13720.0 − 23763.7i 0.625000 − 1.08253i
\(29\) −3978.00 −0.163106 −0.0815532 − 0.996669i \(-0.525988\pi\)
−0.0815532 + 0.996669i \(0.525988\pi\)
\(30\) 0 0
\(31\) −11098.5 − 6407.72i −0.372545 − 0.215089i 0.302025 − 0.953300i \(-0.402338\pi\)
−0.674570 + 0.738211i \(0.735671\pi\)
\(32\) 23040.0 − 39906.5i 0.703125 − 1.21785i
\(33\) 0 0
\(34\) − 36227.6i − 0.921727i
\(35\) 54022.5 − 31189.9i 1.26000 − 0.727461i
\(36\) 0 0
\(37\) 30788.5 + 53327.2i 0.607832 + 1.05280i 0.991597 + 0.129365i \(0.0412938\pi\)
−0.383765 + 0.923431i \(0.625373\pi\)
\(38\) −71442.0 − 41247.1i −1.30198 − 0.751696i
\(39\) 0 0
\(40\) −30240.0 + 17459.1i −0.472500 + 0.272798i
\(41\) 110574.i 1.60436i 0.597082 + 0.802180i \(0.296327\pi\)
−0.597082 + 0.802180i \(0.703673\pi\)
\(42\) 0 0
\(43\) −17414.0 −0.219025 −0.109512 − 0.993985i \(-0.534929\pi\)
−0.109512 + 0.993985i \(0.534929\pi\)
\(44\) 59160.0 + 102468.i 0.694497 + 1.20290i
\(45\) 0 0
\(46\) −35478.0 + 61449.7i −0.364490 + 0.631315i
\(47\) 26554.5 − 15331.2i 0.255767 − 0.147667i −0.366635 − 0.930365i \(-0.619490\pi\)
0.622402 + 0.782698i \(0.286157\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) −209400. −1.67520
\(51\) 0 0
\(52\) −33600.0 − 19399.0i −0.238962 − 0.137965i
\(53\) −30256.5 + 52405.8i −0.203232 + 0.352007i −0.949568 − 0.313562i \(-0.898478\pi\)
0.746336 + 0.665569i \(0.231811\pi\)
\(54\) 0 0
\(55\) 268979.i 1.61670i
\(56\) −65856.0 −0.375000
\(57\) 0 0
\(58\) 23868.0 + 41340.6i 0.122330 + 0.211881i
\(59\) 186826. + 107864.i 0.909667 + 0.525196i 0.880324 − 0.474373i \(-0.157325\pi\)
0.0293430 + 0.999569i \(0.490658\pi\)
\(60\) 0 0
\(61\) 140942. − 81372.6i 0.620940 − 0.358500i −0.156295 − 0.987710i \(-0.549955\pi\)
0.777235 + 0.629211i \(0.216622\pi\)
\(62\) 153785.i 0.645268i
\(63\) 0 0
\(64\) −372736. −1.42188
\(65\) −44100.0 − 76383.4i −0.160583 − 0.278137i
\(66\) 0 0
\(67\) 134388. − 232768.i 0.446825 − 0.773924i −0.551352 − 0.834273i \(-0.685888\pi\)
0.998177 + 0.0603486i \(0.0192212\pi\)
\(68\) −209160. + 120759.i −0.665199 + 0.384053i
\(69\) 0 0
\(70\) −648270. − 374279.i −1.89000 − 1.09119i
\(71\) −101922. −0.284769 −0.142385 − 0.989811i \(-0.545477\pi\)
−0.142385 + 0.989811i \(0.545477\pi\)
\(72\) 0 0
\(73\) 275090. + 158823.i 0.707140 + 0.408267i 0.810001 − 0.586428i \(-0.199466\pi\)
−0.102861 + 0.994696i \(0.532800\pi\)
\(74\) 369462. − 639927.i 0.911748 − 1.57919i
\(75\) 0 0
\(76\) 549961.i 1.25283i
\(77\) −253648. + 439332.i −0.555597 + 0.962323i
\(78\) 0 0
\(79\) −181116. − 313701.i −0.367345 − 0.636261i 0.621804 − 0.783173i \(-0.286400\pi\)
−0.989150 + 0.146912i \(0.953067\pi\)
\(80\) −443520. − 256066.i −0.866250 − 0.500130i
\(81\) 0 0
\(82\) 1.14912e6 − 663445.i 2.08413 − 1.20327i
\(83\) − 216783.i − 0.379133i −0.981868 − 0.189567i \(-0.939292\pi\)
0.981868 − 0.189567i \(-0.0607083\pi\)
\(84\) 0 0
\(85\) −549045. −0.894028
\(86\) 104484. + 180972.i 0.164269 + 0.284521i
\(87\) 0 0
\(88\) 141984. − 245924.i 0.208349 − 0.360871i
\(89\) 1.15577e6 − 667282.i 1.63946 − 0.946541i 0.658437 − 0.752636i \(-0.271218\pi\)
0.981020 − 0.193905i \(-0.0621154\pi\)
\(90\) 0 0
\(91\) − 166346.i − 0.220744i
\(92\) 473040. 0.607483
\(93\) 0 0
\(94\) −318654. − 183975.i −0.383651 − 0.221501i
\(95\) −625118. + 1.08274e6i −0.729106 + 1.26285i
\(96\) 0 0
\(97\) 1.51409e6i 1.65896i 0.558535 + 0.829481i \(0.311364\pi\)
−0.558535 + 0.829481i \(0.688636\pi\)
\(98\) −705894. − 1.22264e6i −0.750000 − 1.29904i
\(99\) 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 63.7.m.a.10.1 2
3.2 odd 2 7.7.d.a.3.1 ✓ 2
7.3 odd 6 441.7.d.a.244.2 2
7.4 even 3 441.7.d.a.244.1 2
7.5 odd 6 inner 63.7.m.a.19.1 2
12.11 even 2 112.7.s.a.17.1 2
21.2 odd 6 49.7.d.b.19.1 2
21.5 even 6 7.7.d.a.5.1 yes 2
21.11 odd 6 49.7.b.a.48.2 2
21.17 even 6 49.7.b.a.48.1 2
21.20 even 2 49.7.d.b.31.1 2
84.47 odd 6 112.7.s.a.33.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.a.3.1 ✓ 2 3.2 odd 2
7.7.d.a.5.1 yes 2 21.5 even 6
49.7.b.a.48.1 2 21.17 even 6
49.7.b.a.48.2 2 21.11 odd 6
49.7.d.b.19.1 2 21.2 odd 6
49.7.d.b.31.1 2 21.20 even 2
63.7.m.a.10.1 2 1.1 even 1 trivial
63.7.m.a.19.1 2 7.5 odd 6 inner
112.7.s.a.17.1 2 12.11 even 2
112.7.s.a.33.1 2 84.47 odd 6
441.7.d.a.244.1 2 7.4 even 3
441.7.d.a.244.2 2 7.3 odd 6