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 33.1
Root \(2.78499i\) of defining polynomial
Character \(\chi\) \(=\) 112.33
Dual form 112.5.s.c.17.1

$q$-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 \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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 −0.519196 + 0.854655i \(0.673769\pi\)
−0.999751 + 0.0223095i \(0.992898\pi\)
\(4\) 0 0
\(5\) −24.9067 − 14.3799i −0.996270 − 0.575197i −0.0891273 − 0.996020i \(-0.528408\pi\)
−0.907142 + 0.420824i \(0.861741\pi\)
\(6\) 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 0.838389 − 0.545073i \(-0.183498\pi\)
−0.891241 + 0.453529i \(0.850165\pi\)
\(12\) 0 0
\(13\) 283.845i 1.67956i 0.542928 + 0.839779i \(0.317316\pi\)
−0.542928 + 0.839779i \(0.682684\pi\)
\(14\) 0 0
\(15\) 453.984 2.01771
\(16\) 0 0
\(17\) −164.093 + 94.7393i −0.567797 + 0.327818i −0.756269 − 0.654261i \(-0.772980\pi\)
0.188472 + 0.982079i \(0.439646\pi\)
\(18\) 0 0
\(19\) 139.870 + 80.7538i 0.387450 + 0.223695i 0.681055 − 0.732232i \(-0.261521\pi\)
−0.293604 + 0.955927i \(0.594855\pi\)
\(20\) 0 0
\(21\) 768.164 + 90.5483i 1.74187 + 0.205325i
\(22\) 0 0
\(23\) 86.2424 − 149.376i 0.163029 − 0.282375i −0.772925 − 0.634498i \(-0.781207\pi\)
0.935954 + 0.352123i \(0.114540\pi\)
\(24\) 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 −0.0577891 − 0.998329i \(-0.518405\pi\)
0.835684 + 0.549211i \(0.185072\pi\)
\(32\) 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 −0.675866 − 0.737025i \(-0.736230\pi\)
0.976215 + 0.216805i \(0.0695635\pi\)
\(38\) 0 0
\(39\) −2240.30 − 3880.31i −1.47291 − 2.55116i
\(40\) 0 0
\(41\) − 2857.61i − 1.69994i −0.526827 − 0.849972i \(-0.676619\pi\)
0.526827 − 0.849972i \(-0.323381\pi\)
\(42\) 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 0.989245 − 0.146270i \(-0.0467268\pi\)
0.367949 + 0.929846i \(0.380060\pi\)
\(48\) 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 0.847173 − 0.531317i \(-0.178303\pi\)
−0.883721 + 0.468015i \(0.844969\pi\)
\(54\) 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 −0.897625 − 0.440760i \(-0.854709\pi\)
−0.0671029 + 0.997746i \(0.521376\pi\)
\(60\) 0 0
\(61\) 1784.80 + 1030.46i 0.479657 + 0.276930i 0.720273 − 0.693690i \(-0.244016\pi\)
−0.240617 + 0.970620i \(0.577350\pi\)
\(62\) 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 0.872972 − 0.487771i \(-0.162190\pi\)
−0.858908 + 0.512130i \(0.828857\pi\)
\(68\) 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 −0.855017 − 0.518600i \(-0.826453\pi\)
0.0216127 + 0.999766i \(0.493120\pi\)
\(74\) 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 −0.954169 − 0.299267i \(-0.903258\pi\)
0.736257 + 0.676702i \(0.236591\pi\)
\(80\) 0 0
\(81\) −4050.17 − 7015.10i −0.617310 − 1.06921i
\(82\) 0 0
\(83\) − 7847.13i − 1.13908i −0.821963 − 0.569541i \(-0.807121\pi\)
0.821963 − 0.569541i \(-0.192879\pi\)
\(84\) 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 0.103319 − 0.994648i \(-0.467054\pi\)
−0.809731 + 0.586801i \(0.800387\pi\)
\(90\) 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 \(0.00320556\pi\)
−0.999949 + 0.0100704i \(0.996794\pi\)
\(98\) 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.33.1 6
4.3 odd 2 28.5.h.a.5.3 ✓ 6
7.2 even 3 784.5.c.e.97.1 6
7.3 odd 6 inner 112.5.s.c.17.1 6
7.5 odd 6 784.5.c.e.97.6 6
12.11 even 2 252.5.z.f.145.2 6
20.3 even 4 700.5.o.a.649.6 12
20.7 even 4 700.5.o.a.649.1 12
20.19 odd 2 700.5.s.a.201.1 6
28.3 even 6 28.5.h.a.17.3 yes 6
28.11 odd 6 196.5.h.c.129.1 6
28.19 even 6 196.5.b.a.97.1 6
28.23 odd 6 196.5.b.a.97.6 6
28.27 even 2 196.5.h.c.117.1 6
84.59 odd 6 252.5.z.f.73.2 6
140.3 odd 12 700.5.o.a.549.1 12
140.59 even 6 700.5.s.a.101.1 6
140.87 odd 12 700.5.o.a.549.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.5.h.a.5.3 ✓ 6 4.3 odd 2
28.5.h.a.17.3 yes 6 28.3 even 6
112.5.s.c.17.1 6 7.3 odd 6 inner
112.5.s.c.33.1 6 1.1 even 1 trivial
196.5.b.a.97.1 6 28.19 even 6
196.5.b.a.97.6 6 28.23 odd 6
196.5.h.c.117.1 6 28.27 even 2
196.5.h.c.129.1 6 28.11 odd 6
252.5.z.f.73.2 6 84.59 odd 6
252.5.z.f.145.2 6 12.11 even 2
700.5.o.a.549.1 12 140.3 odd 12
700.5.o.a.549.6 12 140.87 odd 12
700.5.o.a.649.1 12 20.7 even 4
700.5.o.a.649.6 12 20.3 even 4
700.5.s.a.101.1 6 140.59 even 6
700.5.s.a.201.1 6 20.19 odd 2
784.5.c.e.97.1 6 7.2 even 3
784.5.c.e.97.6 6 7.5 odd 6