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 17.3
Root \(3.17656i\) of defining polynomial
Character \(\chi\) \(=\) 112.17
Dual form 112.5.s.c.33.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(7.81152 + 4.50998i) q^{3} +(-26.9260 + 15.5457i) q^{5} +(48.6720 + 5.65972i) q^{7} +(0.179888 + 0.311574i) q^{9} +(-72.8605 + 126.198i) q^{11} +209.930i q^{13} -280.444 q^{15} +(-162.074 - 93.5735i) q^{17} +(-153.531 + 88.6415i) q^{19} +(354.677 + 263.721i) q^{21} +(166.842 + 288.979i) q^{23} +(170.838 - 295.901i) q^{25} -727.372i q^{27} -1387.57 q^{29} +(1345.74 + 776.966i) q^{31} +(-1138.30 + 657.199i) q^{33} +(-1398.53 + 604.248i) q^{35} +(1138.99 + 1972.78i) q^{37} +(-946.779 + 1639.87i) q^{39} -781.977i q^{41} -837.131 q^{43} +(-9.68729 - 5.59296i) q^{45} +(1296.36 - 748.454i) q^{47} +(2336.94 + 550.940i) q^{49} +(-844.030 - 1461.90i) q^{51} +(2233.55 - 3868.63i) q^{53} -4530.67i q^{55} -1599.09 q^{57} +(3815.50 + 2202.88i) q^{59} +(2257.75 - 1303.51i) q^{61} +(6.99207 + 16.1831i) q^{63} +(-3263.51 - 5652.56i) q^{65} +(2887.15 - 5000.70i) q^{67} +3009.82i q^{69} -1149.20 q^{71} +(-3194.47 - 1844.33i) q^{73} +(2669.01 - 1540.96i) q^{75} +(-4260.52 + 5729.95i) q^{77} +(1349.16 + 2336.81i) q^{79} +(3295.01 - 5707.12i) q^{81} +2689.70i q^{83} +5818.67 q^{85} +(-10839.1 - 6257.93i) q^{87} +(5037.61 - 2908.47i) q^{89} +(-1188.14 + 10217.7i) q^{91} +(7008.21 + 12138.6i) q^{93} +(2755.99 - 4773.51i) q^{95} -7890.42i q^{97} -52.4268 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 27 q^{5} - 66 q^{7} + 90 q^{9} - 135 q^{11} + 486 q^{15} - 1107 q^{17} + 747 q^{19} + 2169 q^{21} - 243 q^{23} + 1878 q^{25} - 540 q^{29} + 5355 q^{31} - 1863 q^{33} - 6021 q^{35} + 2355 q^{37}+ \cdots - 8100 q^{99}+O(q^{100}) \) 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
\(3\) 7.81152 + 4.50998i 0.867947 + 0.501109i 0.866665 − 0.498890i \(-0.166259\pi\)
0.00128125 + 0.999999i \(0.499592\pi\)
\(4\) 0 0
\(5\) −26.9260 + 15.5457i −1.07704 + 0.621829i −0.930096 − 0.367317i \(-0.880277\pi\)
−0.146943 + 0.989145i \(0.546943\pi\)
\(6\) 0 0
\(7\) 48.6720 + 5.65972i 0.993307 + 0.115505i
\(8\) 0 0
\(9\) 0.179888 + 0.311574i 0.00222083 + 0.00384660i
\(10\) 0 0
\(11\) −72.8605 + 126.198i −0.602153 + 1.04296i 0.390341 + 0.920670i \(0.372357\pi\)
−0.992494 + 0.122290i \(0.960976\pi\)
\(12\) 0 0
\(13\) 209.930i 1.24219i 0.783736 + 0.621094i \(0.213311\pi\)
−0.783736 + 0.621094i \(0.786689\pi\)
\(14\) 0 0
\(15\) −280.444 −1.24642
\(16\) 0 0
\(17\) −162.074 − 93.5735i −0.560810 − 0.323784i 0.192661 − 0.981265i \(-0.438288\pi\)
−0.753470 + 0.657482i \(0.771622\pi\)
\(18\) 0 0
\(19\) −153.531 + 88.6415i −0.425295 + 0.245544i −0.697340 − 0.716740i \(-0.745633\pi\)
0.272045 + 0.962284i \(0.412300\pi\)
\(20\) 0 0
\(21\) 354.677 + 263.721i 0.804257 + 0.598007i
\(22\) 0 0
\(23\) 166.842 + 288.979i 0.315392 + 0.546275i 0.979521 − 0.201343i \(-0.0645307\pi\)
−0.664129 + 0.747618i \(0.731197\pi\)
\(24\) 0 0
\(25\) 170.838 − 295.901i 0.273341 − 0.473441i
\(26\) 0 0
\(27\) − 727.372i − 0.997767i
\(28\) 0 0
\(29\) −1387.57 −1.64991 −0.824954 − 0.565199i \(-0.808799\pi\)
−0.824954 + 0.565199i \(0.808799\pi\)
\(30\) 0 0
\(31\) 1345.74 + 776.966i 1.40036 + 0.808497i 0.994429 − 0.105407i \(-0.0336145\pi\)
0.405930 + 0.913904i \(0.366948\pi\)
\(32\) 0 0
\(33\) −1138.30 + 657.199i −1.04527 + 0.603489i
\(34\) 0 0
\(35\) −1398.53 + 604.248i −1.14165 + 0.493264i
\(36\) 0 0
\(37\) 1138.99 + 1972.78i 0.831985 + 1.44104i 0.896462 + 0.443122i \(0.146129\pi\)
−0.0644762 + 0.997919i \(0.520538\pi\)
\(38\) 0 0
\(39\) −946.779 + 1639.87i −0.622471 + 1.07815i
\(40\) 0 0
\(41\) − 781.977i − 0.465185i −0.972574 − 0.232593i \(-0.925279\pi\)
0.972574 − 0.232593i \(-0.0747209\pi\)
\(42\) 0 0
\(43\) −837.131 −0.452748 −0.226374 − 0.974040i \(-0.572687\pi\)
−0.226374 + 0.974040i \(0.572687\pi\)
\(44\) 0 0
\(45\) −9.68729 − 5.59296i −0.00478385 − 0.00276196i
\(46\) 0 0
\(47\) 1296.36 − 748.454i 0.586854 − 0.338820i −0.176999 − 0.984211i \(-0.556639\pi\)
0.763853 + 0.645391i \(0.223305\pi\)
\(48\) 0 0
\(49\) 2336.94 + 550.940i 0.973317 + 0.229463i
\(50\) 0 0
\(51\) −844.030 − 1461.90i −0.324502 − 0.562054i
\(52\) 0 0
\(53\) 2233.55 − 3868.63i 0.795141 − 1.37723i −0.127608 − 0.991825i \(-0.540730\pi\)
0.922749 − 0.385400i \(-0.125937\pi\)
\(54\) 0 0
\(55\) − 4530.67i − 1.49774i
\(56\) 0 0
\(57\) −1599.09 −0.492178
\(58\) 0 0
\(59\) 3815.50 + 2202.88i 1.09609 + 0.632829i 0.935192 − 0.354142i \(-0.115227\pi\)
0.160900 + 0.986971i \(0.448560\pi\)
\(60\) 0 0
\(61\) 2257.75 − 1303.51i 0.606758 − 0.350312i −0.164938 − 0.986304i \(-0.552742\pi\)
0.771695 + 0.635992i \(0.219409\pi\)
\(62\) 0 0
\(63\) 6.99207 + 16.1831i 0.00176167 + 0.00407737i
\(64\) 0 0
\(65\) −3263.51 − 5652.56i −0.772427 − 1.33788i
\(66\) 0 0
\(67\) 2887.15 − 5000.70i 0.643162 − 1.11399i −0.341561 − 0.939860i \(-0.610956\pi\)
0.984723 − 0.174129i \(-0.0557111\pi\)
\(68\) 0 0
\(69\) 3009.82i 0.632183i
\(70\) 0 0
\(71\) −1149.20 −0.227971 −0.113986 − 0.993482i \(-0.536362\pi\)
−0.113986 + 0.993482i \(0.536362\pi\)
\(72\) 0 0
\(73\) −3194.47 − 1844.33i −0.599450 − 0.346093i 0.169375 − 0.985552i \(-0.445825\pi\)
−0.768825 + 0.639459i \(0.779158\pi\)
\(74\) 0 0
\(75\) 2669.01 − 1540.96i 0.474491 − 0.273948i
\(76\) 0 0
\(77\) −4260.52 + 5729.95i −0.718589 + 0.966428i
\(78\) 0 0
\(79\) 1349.16 + 2336.81i 0.216177 + 0.374429i 0.953636 − 0.300963i \(-0.0973079\pi\)
−0.737459 + 0.675392i \(0.763975\pi\)
\(80\) 0 0
\(81\) 3295.01 − 5707.12i 0.502211 − 0.869855i
\(82\) 0 0
\(83\) 2689.70i 0.390433i 0.980760 + 0.195217i \(0.0625410\pi\)
−0.980760 + 0.195217i \(0.937459\pi\)
\(84\) 0 0
\(85\) 5818.67 0.805352
\(86\) 0 0
\(87\) −10839.1 − 6257.93i −1.43203 − 0.826784i
\(88\) 0 0
\(89\) 5037.61 − 2908.47i 0.635982 − 0.367184i −0.147083 − 0.989124i \(-0.546988\pi\)
0.783065 + 0.621940i \(0.213655\pi\)
\(90\) 0 0
\(91\) −1188.14 + 10217.7i −0.143478 + 1.23387i
\(92\) 0 0
\(93\) 7008.21 + 12138.6i 0.810291 + 1.40347i
\(94\) 0 0
\(95\) 2755.99 − 4773.51i 0.305373 − 0.528921i
\(96\) 0 0
\(97\) − 7890.42i − 0.838604i −0.907847 − 0.419302i \(-0.862275\pi\)
0.907847 − 0.419302i \(-0.137725\pi\)
\(98\) 0 0
\(99\) −52.4268 −0.00534913
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.17.3 6
4.3 odd 2 28.5.h.a.17.1 yes 6
7.3 odd 6 784.5.c.e.97.5 6
7.4 even 3 784.5.c.e.97.2 6
7.5 odd 6 inner 112.5.s.c.33.3 6
12.11 even 2 252.5.z.f.73.3 6
20.3 even 4 700.5.o.a.549.5 12
20.7 even 4 700.5.o.a.549.2 12
20.19 odd 2 700.5.s.a.101.3 6
28.3 even 6 196.5.b.a.97.2 6
28.11 odd 6 196.5.b.a.97.5 6
28.19 even 6 28.5.h.a.5.1 ✓ 6
28.23 odd 6 196.5.h.c.117.3 6
28.27 even 2 196.5.h.c.129.3 6
84.47 odd 6 252.5.z.f.145.3 6
140.19 even 6 700.5.s.a.201.3 6
140.47 odd 12 700.5.o.a.649.5 12
140.103 odd 12 700.5.o.a.649.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.5.h.a.5.1 ✓ 6 28.19 even 6
28.5.h.a.17.1 yes 6 4.3 odd 2
112.5.s.c.17.3 6 1.1 even 1 trivial
112.5.s.c.33.3 6 7.5 odd 6 inner
196.5.b.a.97.2 6 28.3 even 6
196.5.b.a.97.5 6 28.11 odd 6
196.5.h.c.117.3 6 28.23 odd 6
196.5.h.c.129.3 6 28.27 even 2
252.5.z.f.73.3 6 12.11 even 2
252.5.z.f.145.3 6 84.47 odd 6
700.5.o.a.549.2 12 20.7 even 4
700.5.o.a.549.5 12 20.3 even 4
700.5.o.a.649.2 12 140.103 odd 12
700.5.o.a.649.5 12 140.47 odd 12
700.5.s.a.101.3 6 20.19 odd 2
700.5.s.a.201.3 6 140.19 even 6
784.5.c.e.97.2 6 7.4 even 3
784.5.c.e.97.5 6 7.3 odd 6