Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(17,112)] 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("112.17"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.11337408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 18x^{4} + 81x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.1
Root \(-2.78499i\) of defining polynomial
Character \(\chi\) \(=\) 112.17
Dual form 112.5.s.c.33.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-13.6705 - 7.89268i) q^{3} +(-24.9067 + 14.3799i) q^{5} +(-39.2754 + 29.2993i) q^{7} +(84.0888 + 145.646i) q^{9} +(-6.39517 + 11.0768i) q^{11} -283.845i q^{13} +453.984 q^{15} +(-164.093 - 94.7393i) q^{17} +(139.870 - 80.7538i) q^{19} +(768.164 - 90.5483i) q^{21} +(86.2424 + 149.376i) q^{23} +(101.064 - 175.048i) q^{25} -1376.13i q^{27} +711.049 q^{29} +(747.557 + 431.602i) q^{31} +(174.851 - 100.950i) q^{33} +(556.901 - 1294.53i) q^{35} +(411.178 + 712.181i) q^{37} +(-2240.30 + 3880.31i) q^{39} +2857.61i q^{41} -1325.55 q^{43} +(-4188.76 - 2418.38i) q^{45} +(2998.04 - 1730.92i) q^{47} +(684.107 - 2301.48i) q^{49} +(1495.49 + 2590.27i) q^{51} +(-102.663 + 177.817i) q^{53} -367.848i q^{55} -2549.46 q^{57} +(-3358.22 - 1938.87i) q^{59} +(1784.80 - 1030.46i) q^{61} +(-7569.94 - 3256.56i) q^{63} +(4081.67 + 7069.66i) q^{65} +(63.1323 - 109.348i) q^{67} -2722.73i q^{69} +7590.56 q^{71} +(-4441.21 - 2564.13i) q^{73} +(-2763.19 + 1595.33i) q^{75} +(-73.3682 - 622.417i) q^{77} +(-1359.99 - 2355.57i) q^{79} +(-4050.17 + 7015.10i) q^{81} +7847.13i q^{83} +5449.37 q^{85} +(-9720.41 - 5612.08i) q^{87} +(-5595.49 + 3230.56i) q^{89} +(8316.46 + 11148.1i) q^{91} +(-6812.99 - 11800.5i) q^{93} +(-2322.46 + 4022.63i) q^{95} -189.505i q^{97} -2151.05 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 27 q^{5} - 66 q^{7} + 90 q^{9} - 135 q^{11} + 486 q^{15} - 1107 q^{17} + 747 q^{19} + 2169 q^{21} - 243 q^{23} + 1878 q^{25} - 540 q^{29} + 5355 q^{31} - 1863 q^{33} - 6021 q^{35} + 2355 q^{37}+ \cdots - 8100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −13.6705 − 7.89268i −1.51895 − 0.876965i −0.999751 − 0.0223095i \(-0.992898\pi\)
−0.519196 − 0.854655i \(-0.673769\pi\)
\(4\) 0 0
\(5\) −24.9067 + 14.3799i −0.996270 + 0.575197i −0.907142 − 0.420824i \(-0.861741\pi\)
−0.0891273 + 0.996020i \(0.528408\pi\)
\(6\) 0 0
\(7\) −39.2754 + 29.2993i −0.801538 + 0.597944i
\(8\) 0 0
\(9\) 84.0888 + 145.646i 1.03813 + 1.79810i
\(10\) 0 0
\(11\) −6.39517 + 11.0768i −0.0528526 + 0.0915434i −0.891241 − 0.453529i \(-0.850165\pi\)
0.838389 + 0.545073i \(0.183498\pi\)
\(12\) 0 0
\(13\) − 283.845i − 1.67956i −0.542928 − 0.839779i \(-0.682684\pi\)
0.542928 − 0.839779i \(-0.317316\pi\)
\(14\) 0 0
\(15\) 453.984 2.01771
\(16\) 0 0
\(17\) −164.093 − 94.7393i −0.567797 − 0.327818i 0.188472 − 0.982079i \(-0.439646\pi\)
−0.756269 + 0.654261i \(0.772980\pi\)
\(18\) 0 0
\(19\) 139.870 − 80.7538i 0.387450 − 0.223695i −0.293604 − 0.955927i \(-0.594855\pi\)
0.681055 + 0.732232i \(0.261521\pi\)
\(20\) 0 0
\(21\) 768.164 − 90.5483i 1.74187 − 0.205325i
\(22\) 0 0
\(23\) 86.2424 + 149.376i 0.163029 + 0.282375i 0.935954 − 0.352123i \(-0.114540\pi\)
−0.772925 + 0.634498i \(0.781207\pi\)
\(24\) 0 0
\(25\) 101.064 − 175.048i 0.161702 − 0.280077i
\(26\) 0 0
\(27\) − 1376.13i − 1.88770i
\(28\) 0 0
\(29\) 711.049 0.845480 0.422740 − 0.906251i \(-0.361068\pi\)
0.422740 + 0.906251i \(0.361068\pi\)
\(30\) 0 0
\(31\) 747.557 + 431.602i 0.777894 + 0.449118i 0.835684 − 0.549211i \(-0.185072\pi\)
−0.0577891 + 0.998329i \(0.518405\pi\)
\(32\) 0 0
\(33\) 174.851 − 100.950i 0.160561 − 0.0926998i
\(34\) 0 0
\(35\) 556.901 − 1294.53i 0.454613 − 1.05676i
\(36\) 0 0
\(37\) 411.178 + 712.181i 0.300349 + 0.520220i 0.976215 − 0.216805i \(-0.0695635\pi\)
−0.675866 + 0.737025i \(0.736230\pi\)
\(38\) 0 0
\(39\) −2240.30 + 3880.31i −1.47291 + 2.55116i
\(40\) 0 0
\(41\) 2857.61i 1.69994i 0.526827 + 0.849972i \(0.323381\pi\)
−0.526827 + 0.849972i \(0.676619\pi\)
\(42\) 0 0
\(43\) −1325.55 −0.716902 −0.358451 − 0.933548i \(-0.616695\pi\)
−0.358451 + 0.933548i \(0.616695\pi\)
\(44\) 0 0
\(45\) −4188.76 − 2418.38i −2.06852 − 1.19426i
\(46\) 0 0
\(47\) 2998.04 − 1730.92i 1.35719 − 0.783576i 0.367949 − 0.929846i \(-0.380060\pi\)
0.989245 + 0.146270i \(0.0467268\pi\)
\(48\) 0 0
\(49\) 684.107 − 2301.48i 0.284926 − 0.958549i
\(50\) 0 0
\(51\) 1495.49 + 2590.27i 0.574969 + 0.995875i
\(52\) 0 0
\(53\) −102.663 + 177.817i −0.0365478 + 0.0633027i −0.883721 − 0.468015i \(-0.844969\pi\)
0.847173 + 0.531317i \(0.178303\pi\)
\(54\) 0 0
\(55\) − 367.848i − 0.121603i
\(56\) 0 0
\(57\) −2549.46 −0.784689
\(58\) 0 0
\(59\) −3358.22 − 1938.87i −0.964728 − 0.556986i −0.0671029 − 0.997746i \(-0.521376\pi\)
−0.897625 + 0.440760i \(0.854709\pi\)
\(60\) 0 0
\(61\) 1784.80 − 1030.46i 0.479657 − 0.276930i −0.240617 − 0.970620i \(-0.577350\pi\)
0.720273 + 0.693690i \(0.244016\pi\)
\(62\) 0 0
\(63\) −7569.94 − 3256.56i −1.90727 − 0.820500i
\(64\) 0 0
\(65\) 4081.67 + 7069.66i 0.966076 + 1.67329i
\(66\) 0 0
\(67\) 63.1323 − 109.348i 0.0140638 − 0.0243592i −0.858908 − 0.512130i \(-0.828857\pi\)
0.872972 + 0.487771i \(0.162190\pi\)
\(68\) 0 0
\(69\) − 2722.73i − 0.571883i
\(70\) 0 0
\(71\) 7590.56 1.50577 0.752883 − 0.658155i \(-0.228663\pi\)
0.752883 + 0.658155i \(0.228663\pi\)
\(72\) 0 0
\(73\) −4441.21 − 2564.13i −0.833404 − 0.481166i 0.0216127 − 0.999766i \(-0.493120\pi\)
−0.855017 + 0.518600i \(0.826453\pi\)
\(74\) 0 0
\(75\) −2763.19 + 1595.33i −0.491235 + 0.283614i
\(76\) 0 0
\(77\) −73.3682 − 622.417i −0.0123745 − 0.104978i
\(78\) 0 0
\(79\) −1359.99 − 2355.57i −0.217912 − 0.377435i 0.736257 − 0.676702i \(-0.236591\pi\)
−0.954169 + 0.299267i \(0.903258\pi\)
\(80\) 0 0
\(81\) −4050.17 + 7015.10i −0.617310 + 1.06921i
\(82\) 0 0
\(83\) 7847.13i 1.13908i 0.821963 + 0.569541i \(0.192879\pi\)
−0.821963 + 0.569541i \(0.807121\pi\)
\(84\) 0 0
\(85\) 5449.37 0.754238
\(86\) 0 0
\(87\) −9720.41 − 5612.08i −1.28424 − 0.741456i
\(88\) 0 0
\(89\) −5595.49 + 3230.56i −0.706412 + 0.407847i −0.809731 − 0.586801i \(-0.800387\pi\)
0.103319 + 0.994648i \(0.467054\pi\)
\(90\) 0 0
\(91\) 8316.46 + 11148.1i 1.00428 + 1.34623i
\(92\) 0 0
\(93\) −6812.99 − 11800.5i −0.787720 − 1.36437i
\(94\) 0 0
\(95\) −2322.46 + 4022.63i −0.257337 + 0.445720i
\(96\) 0 0
\(97\) − 189.505i − 0.0201408i −0.999949 − 0.0100704i \(-0.996794\pi\)
0.999949 − 0.0100704i \(-0.00320556\pi\)
\(98\) 0 0
\(99\) −2151.05 −0.219472
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 112.5.s.c.17.1 6
4.3 odd 2 28.5.h.a.17.3 yes 6
7.3 odd 6 784.5.c.e.97.1 6
7.4 even 3 784.5.c.e.97.6 6
7.5 odd 6 inner 112.5.s.c.33.1 6
12.11 even 2 252.5.z.f.73.2 6
20.3 even 4 700.5.o.a.549.1 12
20.7 even 4 700.5.o.a.549.6 12
20.19 odd 2 700.5.s.a.101.1 6
28.3 even 6 196.5.b.a.97.6 6
28.11 odd 6 196.5.b.a.97.1 6
28.19 even 6 28.5.h.a.5.3 ✓ 6
28.23 odd 6 196.5.h.c.117.1 6
28.27 even 2 196.5.h.c.129.1 6
84.47 odd 6 252.5.z.f.145.2 6
140.19 even 6 700.5.s.a.201.1 6
140.47 odd 12 700.5.o.a.649.1 12
140.103 odd 12 700.5.o.a.649.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.5.h.a.5.3 ✓ 6 28.19 even 6
28.5.h.a.17.3 yes 6 4.3 odd 2
112.5.s.c.17.1 6 1.1 even 1 trivial
112.5.s.c.33.1 6 7.5 odd 6 inner
196.5.b.a.97.1 6 28.11 odd 6
196.5.b.a.97.6 6 28.3 even 6
196.5.h.c.117.1 6 28.23 odd 6
196.5.h.c.129.1 6 28.27 even 2
252.5.z.f.73.2 6 12.11 even 2
252.5.z.f.145.2 6 84.47 odd 6
700.5.o.a.549.1 12 20.3 even 4
700.5.o.a.549.6 12 20.7 even 4
700.5.o.a.649.1 12 140.47 odd 12
700.5.o.a.649.6 12 140.103 odd 12
700.5.s.a.101.1 6 20.19 odd 2
700.5.s.a.201.1 6 140.19 even 6
784.5.c.e.97.1 6 7.3 odd 6
784.5.c.e.97.6 6 7.4 even 3