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 17.2
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 112.17
Dual form 112.7.s.b.33.2

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

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

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
\(3\) 23.0772 + 13.3236i 0.854710 + 0.493467i 0.862237 − 0.506505i \(-0.169063\pi\)
−0.00752738 + 0.999972i \(0.502396\pi\)
\(4\) 0 0
\(5\) 68.5660 − 39.5866i 0.548528 − 0.316693i −0.200000 − 0.979796i \(-0.564094\pi\)
0.748528 + 0.663103i \(0.230761\pi\)
\(6\) 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 0.343025 + 0.939326i \(0.388548\pi\)
−0.984993 + 0.172594i \(0.944785\pi\)
\(12\) 0 0
\(13\) 3129.09i 1.42426i 0.702049 + 0.712129i \(0.252269\pi\)
−0.702049 + 0.712129i \(0.747731\pi\)
\(14\) 0 0
\(15\) 2109.75 0.625110
\(16\) 0 0
\(17\) 3529.96 + 2038.03i 0.718494 + 0.414823i 0.814198 − 0.580587i \(-0.197177\pi\)
−0.0957039 + 0.995410i \(0.530510\pi\)
\(18\) 0 0
\(19\) −5085.89 + 2936.34i −0.741492 + 0.428101i −0.822611 − 0.568604i \(-0.807484\pi\)
0.0811196 + 0.996704i \(0.474150\pi\)
\(20\) 0 0
\(21\) −6952.95 − 5932.62i −0.750778 − 0.640602i
\(22\) 0 0
\(23\) 6660.75 + 11536.8i 0.547444 + 0.948201i 0.998449 + 0.0556791i \(0.0177324\pi\)
−0.451005 + 0.892522i \(0.648934\pi\)
\(24\) 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 0.664090 − 0.747653i \(-0.268819\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(32\) 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 0.888022 − 0.459802i \(-0.152080\pi\)
−0.842211 + 0.539148i \(0.818746\pi\)
\(38\) 0 0
\(39\) −41690.8 + 72210.6i −0.702824 + 1.21733i
\(40\) 0 0
\(41\) 19308.8i 0.280158i 0.990140 + 0.140079i \(0.0447357\pi\)
−0.990140 + 0.140079i \(0.955264\pi\)
\(42\) 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 −0.206592 − 0.978427i \(-0.566237\pi\)
0.744047 + 0.668128i \(0.232904\pi\)
\(48\) 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 0.497568 + 0.867425i \(0.334226\pi\)
−0.999996 + 0.00280549i \(0.999107\pi\)
\(54\) 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 0.365773 − 0.930704i \(-0.380805\pi\)
−0.623127 + 0.782121i \(0.714138\pi\)
\(60\) 0 0
\(61\) −85403.4 + 49307.7i −0.376258 + 0.217233i −0.676189 − 0.736728i \(-0.736370\pi\)
0.299931 + 0.953961i \(0.403036\pi\)
\(62\) 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 0.481403 + 0.876499i \(0.340127\pi\)
−0.999772 + 0.0213423i \(0.993206\pi\)
\(68\) 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 −0.496912 − 0.867801i \(-0.665533\pi\)
−0.999994 + 0.00356186i \(0.998866\pi\)
\(74\) 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 0.656774 − 0.754087i \(-0.271921\pi\)
−0.981446 + 0.191739i \(0.938587\pi\)
\(80\) 0 0
\(81\) 258643. − 447983.i 0.486682 − 0.842958i
\(82\) 0 0
\(83\) − 832356.i − 1.45571i −0.685731 − 0.727855i \(-0.740517\pi\)
0.685731 − 0.727855i \(-0.259483\pi\)
\(84\) 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 −0.248450 − 0.968645i \(-0.579921\pi\)
0.714646 + 0.699486i \(0.246588\pi\)
\(90\) 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 \(-0.803814\pi\)
0.816001 − 0.578050i \(-0.196186\pi\)
\(98\) 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.17.2 4
4.3 odd 2 7.7.d.b.3.1 ✓ 4
7.5 odd 6 inner 112.7.s.b.33.2 4
12.11 even 2 63.7.m.b.10.2 4
28.3 even 6 49.7.b.b.48.3 4
28.11 odd 6 49.7.b.b.48.4 4
28.19 even 6 7.7.d.b.5.1 yes 4
28.23 odd 6 49.7.d.c.19.1 4
28.27 even 2 49.7.d.c.31.1 4
84.11 even 6 441.7.d.b.244.1 4
84.47 odd 6 63.7.m.b.19.2 4
84.59 odd 6 441.7.d.b.244.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.b.3.1 ✓ 4 4.3 odd 2
7.7.d.b.5.1 yes 4 28.19 even 6
49.7.b.b.48.3 4 28.3 even 6
49.7.b.b.48.4 4 28.11 odd 6
49.7.d.c.19.1 4 28.23 odd 6
49.7.d.c.31.1 4 28.27 even 2
63.7.m.b.10.2 4 12.11 even 2
63.7.m.b.19.2 4 84.47 odd 6
112.7.s.b.17.2 4 1.1 even 1 trivial
112.7.s.b.33.2 4 7.5 odd 6 inner
441.7.d.b.244.1 4 84.11 even 6
441.7.d.b.244.2 4 84.59 odd 6