Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,7,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, 7); 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, 7, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 112.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-18,0,-150] 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: \(25.7660573654\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 33.2
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 112.33
Dual form 112.7.s.b.17.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(23.0772 - 13.3236i) q^{3} +(68.5660 + 39.5866i) q^{5} +(-337.286 + 62.3451i) q^{7} +(-9.46299 + 16.3904i) q^{9} +(-854.459 - 1479.97i) q^{11} -3129.09i q^{13} +2109.75 q^{15} +(3529.96 - 2038.03i) q^{17} +(-5085.89 - 2936.34i) q^{19} +(-6952.95 + 5932.62i) q^{21} +(6660.75 - 11536.8i) q^{23} +(-4678.30 - 8103.05i) q^{25} +19930.1i q^{27} +6510.23 q^{29} +(10386.6 - 5996.69i) q^{31} +(-39437.0 - 22769.0i) q^{33} +(-25594.4 - 9077.27i) q^{35} +(2320.45 - 4019.14i) q^{37} +(-41690.8 - 72210.6i) q^{39} -19308.8i q^{41} -91636.4 q^{43} +(-1297.68 + 749.215i) q^{45} +(55800.2 + 32216.2i) q^{47} +(109875. - 42056.3i) q^{49} +(54307.7 - 94063.7i) q^{51} +(-74799.9 - 129557. i) q^{53} -135301. i q^{55} -156491. q^{57} +(-52855.0 + 30515.9i) q^{59} +(-85403.4 - 49307.7i) q^{61} +(2169.88 - 6118.22i) q^{63} +(123870. - 214549. i) q^{65} +(-155906. - 270038. i) q^{67} -354981. i q^{69} +401209. q^{71} +(-582322. + 336204. i) q^{73} +(-215924. - 124664. i) q^{75} +(380466. + 445901. i) q^{77} +(-160076. + 277260. i) q^{79} +(258643. + 447983. i) q^{81} +832356. i q^{83} +322714. q^{85} +(150238. - 86739.7i) q^{87} +(328654. + 189748. i) q^{89} +(195084. + 1.05540e6i) q^{91} +(159795. - 276773. i) q^{93} +(-232480. - 402666. i) q^{95} +1.05514e6i q^{97} +32342.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} - 150 q^{5} - 280 q^{7} + 624 q^{9} - 1882 q^{11} + 16500 q^{15} + 13458 q^{17} - 18078 q^{19} - 16170 q^{21} - 2470 q^{23} + 2500 q^{25} - 34544 q^{29} + 17202 q^{31} - 67770 q^{33} - 154350 q^{35}+ \cdots - 157872 q^{99}+O(q^{100}) \) 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
\(3\) 23.0772 − 13.3236i 0.854710 − 0.493467i −0.00752738 − 0.999972i \(-0.502396\pi\)
0.862237 + 0.506505i \(0.169063\pi\)
\(4\) 0 0
\(5\) 68.5660 + 39.5866i 0.548528 + 0.316693i 0.748528 − 0.663103i \(-0.230761\pi\)
−0.200000 + 0.979796i \(0.564094\pi\)
\(6\) 0 0
\(7\) −337.286 + 62.3451i −0.983342 + 0.181764i
\(8\) 0 0
\(9\) −9.46299 + 16.3904i −0.0129808 + 0.0224834i
\(10\) 0 0
\(11\) −854.459 − 1479.97i −0.641968 − 1.11192i −0.984993 − 0.172594i \(-0.944785\pi\)
0.343025 − 0.939326i \(-0.388548\pi\)
\(12\) 0 0
\(13\) − 3129.09i − 1.42426i −0.702049 − 0.712129i \(-0.747731\pi\)
0.702049 − 0.712129i \(-0.252269\pi\)
\(14\) 0 0
\(15\) 2109.75 0.625110
\(16\) 0 0
\(17\) 3529.96 − 2038.03i 0.718494 − 0.414823i −0.0957039 − 0.995410i \(-0.530510\pi\)
0.814198 + 0.580587i \(0.197177\pi\)
\(18\) 0 0
\(19\) −5085.89 − 2936.34i −0.741492 − 0.428101i 0.0811196 − 0.996704i \(-0.474150\pi\)
−0.822611 + 0.568604i \(0.807484\pi\)
\(20\) 0 0
\(21\) −6952.95 + 5932.62i −0.750778 + 0.640602i
\(22\) 0 0
\(23\) 6660.75 − 11536.8i 0.547444 − 0.948201i −0.451005 − 0.892522i \(-0.648934\pi\)
0.998449 − 0.0556791i \(-0.0177324\pi\)
\(24\) 0 0
\(25\) −4678.30 − 8103.05i −0.299411 − 0.518596i
\(26\) 0 0
\(27\) 19930.1i 1.01256i
\(28\) 0 0
\(29\) 6510.23 0.266933 0.133466 − 0.991053i \(-0.457389\pi\)
0.133466 + 0.991053i \(0.457389\pi\)
\(30\) 0 0
\(31\) 10386.6 − 5996.69i 0.348648 − 0.201292i −0.315442 − 0.948945i \(-0.602153\pi\)
0.664090 + 0.747653i \(0.268819\pi\)
\(32\) 0 0
\(33\) −39437.0 − 22769.0i −1.09739 − 0.633580i
\(34\) 0 0
\(35\) −25594.4 − 9077.27i −0.596954 − 0.211715i
\(36\) 0 0
\(37\) 2320.45 − 4019.14i 0.0458107 − 0.0793465i −0.842211 − 0.539148i \(-0.818746\pi\)
0.888022 + 0.459802i \(0.152080\pi\)
\(38\) 0 0
\(39\) −41690.8 − 72210.6i −0.702824 − 1.21733i
\(40\) 0 0
\(41\) − 19308.8i − 0.280158i −0.990140 − 0.140079i \(-0.955264\pi\)
0.990140 − 0.140079i \(-0.0447357\pi\)
\(42\) 0 0
\(43\) −91636.4 −1.15256 −0.576279 − 0.817253i \(-0.695496\pi\)
−0.576279 + 0.817253i \(0.695496\pi\)
\(44\) 0 0
\(45\) −1297.68 + 749.215i −0.0142406 + 0.00822184i
\(46\) 0 0
\(47\) 55800.2 + 32216.2i 0.537455 + 0.310300i 0.744047 − 0.668128i \(-0.232904\pi\)
−0.206592 + 0.978427i \(0.566237\pi\)
\(48\) 0 0
\(49\) 109875. − 42056.3i 0.933924 − 0.357473i
\(50\) 0 0
\(51\) 54307.7 − 94063.7i 0.409403 − 0.709106i
\(52\) 0 0
\(53\) −74799.9 − 129557.i −0.502428 − 0.870230i −0.999996 − 0.00280549i \(-0.999107\pi\)
0.497568 − 0.867425i \(-0.334226\pi\)
\(54\) 0 0
\(55\) − 135301.i − 0.813226i
\(56\) 0 0
\(57\) −156491. −0.845014
\(58\) 0 0
\(59\) −52855.0 + 30515.9i −0.257354 + 0.148583i −0.623127 − 0.782121i \(-0.714138\pi\)
0.365773 + 0.930704i \(0.380805\pi\)
\(60\) 0 0
\(61\) −85403.4 − 49307.7i −0.376258 − 0.217233i 0.299931 − 0.953961i \(-0.403036\pi\)
−0.676189 + 0.736728i \(0.736370\pi\)
\(62\) 0 0
\(63\) 2169.88 − 6118.22i 0.00867788 − 0.0244683i
\(64\) 0 0
\(65\) 123870. − 214549.i 0.451052 − 0.781245i
\(66\) 0 0
\(67\) −155906. − 270038.i −0.518369 − 0.897842i −0.999772 − 0.0213423i \(-0.993206\pi\)
0.481403 − 0.876499i \(-0.340127\pi\)
\(68\) 0 0
\(69\) − 354981.i − 1.08058i
\(70\) 0 0
\(71\) 401209. 1.12097 0.560487 − 0.828163i \(-0.310614\pi\)
0.560487 + 0.828163i \(0.310614\pi\)
\(72\) 0 0
\(73\) −582322. + 336204.i −1.49691 + 0.864239i −0.999994 − 0.00356186i \(-0.998866\pi\)
−0.496912 + 0.867801i \(0.665533\pi\)
\(74\) 0 0
\(75\) −215924. − 124664.i −0.511819 − 0.295499i
\(76\) 0 0
\(77\) 380466. + 445901.i 0.833381 + 0.976712i
\(78\) 0 0
\(79\) −160076. + 277260.i −0.324672 + 0.562348i −0.981446 − 0.191739i \(-0.938587\pi\)
0.656774 + 0.754087i \(0.271921\pi\)
\(80\) 0 0
\(81\) 258643. + 447983.i 0.486682 + 0.842958i
\(82\) 0 0
\(83\) 832356.i 1.45571i 0.685731 + 0.727855i \(0.259483\pi\)
−0.685731 + 0.727855i \(0.740517\pi\)
\(84\) 0 0
\(85\) 322714. 0.525486
\(86\) 0 0
\(87\) 150238. − 86739.7i 0.228150 − 0.131723i
\(88\) 0 0
\(89\) 328654. + 189748.i 0.466196 + 0.269158i 0.714646 − 0.699486i \(-0.246588\pi\)
−0.248450 + 0.968645i \(0.579921\pi\)
\(90\) 0 0
\(91\) 195084. + 1.05540e6i 0.258879 + 1.40053i
\(92\) 0 0
\(93\) 159795. − 276773.i 0.198662 − 0.344092i
\(94\) 0 0
\(95\) −232480. − 402666.i −0.271153 − 0.469650i
\(96\) 0 0
\(97\) 1.05514e6i 1.15610i 0.816001 + 0.578050i \(0.196186\pi\)
−0.816001 + 0.578050i \(0.803814\pi\)
\(98\) 0 0
\(99\) 32342.9 0.0333330
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 112.7.s.b.33.2 4
4.3 odd 2 7.7.d.b.5.1 yes 4
7.3 odd 6 inner 112.7.s.b.17.2 4
12.11 even 2 63.7.m.b.19.2 4
28.3 even 6 7.7.d.b.3.1 ✓ 4
28.11 odd 6 49.7.d.c.31.1 4
28.19 even 6 49.7.b.b.48.4 4
28.23 odd 6 49.7.b.b.48.3 4
28.27 even 2 49.7.d.c.19.1 4
84.23 even 6 441.7.d.b.244.2 4
84.47 odd 6 441.7.d.b.244.1 4
84.59 odd 6 63.7.m.b.10.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.b.3.1 ✓ 4 28.3 even 6
7.7.d.b.5.1 yes 4 4.3 odd 2
49.7.b.b.48.3 4 28.23 odd 6
49.7.b.b.48.4 4 28.19 even 6
49.7.d.c.19.1 4 28.27 even 2
49.7.d.c.31.1 4 28.11 odd 6
63.7.m.b.10.2 4 84.59 odd 6
63.7.m.b.19.2 4 12.11 even 2
112.7.s.b.17.2 4 7.3 odd 6 inner
112.7.s.b.33.2 4 1.1 even 1 trivial
441.7.d.b.244.1 4 84.47 odd 6
441.7.d.b.244.2 4 84.23 even 6