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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(79,112)] 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("112.79"); 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([3, 0, 2])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,9] 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} + 485 x^{8} - 1910 x^{7} + 81837 x^{6} - 238847 x^{5} + 5758115 x^{4} + \cdots + 1406445775 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 95.3
Root \(0.500000 + 3.92097i\) of defining polynomial
Character \(\chi\) \(=\) 112.95
Dual form 112.5.r.b.79.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(4.14566 + 2.39350i) q^{3} +(-1.24690 - 2.15969i) q^{5} +(-23.3384 - 43.0850i) q^{7} +(-29.0423 - 50.3028i) q^{9} +(-59.9985 - 34.6401i) q^{11} +67.4216 q^{13} -11.9378i q^{15} +(236.065 - 408.877i) q^{17} +(-110.243 + 63.6489i) q^{19} +(6.37111 - 234.476i) q^{21} +(-472.122 + 272.580i) q^{23} +(309.390 - 535.880i) q^{25} -665.798i q^{27} +506.253 q^{29} +(-529.390 - 305.644i) q^{31} +(-165.822 - 287.212i) q^{33} +(-63.9498 + 104.126i) q^{35} +(-688.583 - 1192.66i) q^{37} +(279.507 + 161.373i) q^{39} +963.072 q^{41} -395.563i q^{43} +(-72.4256 + 125.445i) q^{45} +(-722.295 + 417.017i) q^{47} +(-1311.64 + 2011.07i) q^{49} +(1957.29 - 1130.04i) q^{51} +(-1238.73 + 2145.55i) q^{53} +172.771i q^{55} -609.374 q^{57} +(1185.48 + 684.435i) q^{59} +(2147.71 + 3719.94i) q^{61} +(-1489.50 + 2425.27i) q^{63} +(-84.0678 - 145.610i) q^{65} +(6589.71 + 3804.57i) q^{67} -2609.67 q^{69} +2869.42i q^{71} +(-1728.76 + 2994.30i) q^{73} +(2565.26 - 1481.05i) q^{75} +(-92.2064 + 3393.48i) q^{77} +(-3969.87 + 2292.00i) q^{79} +(-758.843 + 1314.36i) q^{81} -12450.4i q^{83} -1177.40 q^{85} +(2098.75 + 1211.71i) q^{87} +(1412.52 + 2446.55i) q^{89} +(-1573.51 - 2904.86i) q^{91} +(-1463.11 - 2534.19i) q^{93} +(274.924 + 158.727i) q^{95} -6275.53 q^{97} +4024.12i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 9 q^{3} + 9 q^{5} + 10 q^{7} + 74 q^{9} + 189 q^{11} + 84 q^{13} - 435 q^{17} + 357 q^{19} - 61 q^{21} + 1269 q^{23} - 776 q^{25} + 660 q^{29} + 969 q^{31} + 1759 q^{33} - 1521 q^{35} - 583 q^{37}+ \cdots - 43356 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 4.14566 + 2.39350i 0.460629 + 0.265944i 0.712309 − 0.701866i \(-0.247650\pi\)
−0.251680 + 0.967811i \(0.580983\pi\)
\(4\) 0 0
\(5\) −1.24690 − 2.15969i −0.0498759 − 0.0863876i 0.840010 − 0.542571i \(-0.182549\pi\)
−0.889886 + 0.456184i \(0.849216\pi\)
\(6\) 0 0
\(7\) −23.3384 − 43.0850i −0.476293 − 0.879287i
\(8\) 0 0
\(9\) −29.0423 − 50.3028i −0.358547 − 0.621022i
\(10\) 0 0
\(11\) −59.9985 − 34.6401i −0.495855 − 0.286282i 0.231145 − 0.972919i \(-0.425753\pi\)
−0.727000 + 0.686637i \(0.759086\pi\)
\(12\) 0 0
\(13\) 67.4216 0.398944 0.199472 − 0.979903i \(-0.436077\pi\)
0.199472 + 0.979903i \(0.436077\pi\)
\(14\) 0 0
\(15\) − 11.9378i − 0.0530568i
\(16\) 0 0
\(17\) 236.065 − 408.877i 0.816834 − 1.41480i −0.0911692 − 0.995835i \(-0.529060\pi\)
0.908003 − 0.418963i \(-0.137606\pi\)
\(18\) 0 0
\(19\) −110.243 + 63.6489i −0.305383 + 0.176313i −0.644858 − 0.764302i \(-0.723084\pi\)
0.339476 + 0.940615i \(0.389750\pi\)
\(20\) 0 0
\(21\) 6.37111 − 234.476i 0.0144470 − 0.531692i
\(22\) 0 0
\(23\) −472.122 + 272.580i −0.892479 + 0.515273i −0.874753 − 0.484570i \(-0.838976\pi\)
−0.0177267 + 0.999843i \(0.505643\pi\)
\(24\) 0 0
\(25\) 309.390 − 535.880i 0.495025 − 0.857408i
\(26\) 0 0
\(27\) − 665.798i − 0.913303i
\(28\) 0 0
\(29\) 506.253 0.601965 0.300983 − 0.953630i \(-0.402685\pi\)
0.300983 + 0.953630i \(0.402685\pi\)
\(30\) 0 0
\(31\) −529.390 − 305.644i −0.550874 − 0.318047i 0.198600 − 0.980081i \(-0.436360\pi\)
−0.749475 + 0.662033i \(0.769694\pi\)
\(32\) 0 0
\(33\) −165.822 − 287.212i −0.152270 − 0.263740i
\(34\) 0 0
\(35\) −63.9498 + 104.126i −0.0522039 + 0.0850010i
\(36\) 0 0
\(37\) −688.583 − 1192.66i −0.502983 − 0.871192i −0.999994 − 0.00344766i \(-0.998903\pi\)
0.497011 − 0.867744i \(-0.334431\pi\)
\(38\) 0 0
\(39\) 279.507 + 161.373i 0.183765 + 0.106097i
\(40\) 0 0
\(41\) 963.072 0.572916 0.286458 − 0.958093i \(-0.407522\pi\)
0.286458 + 0.958093i \(0.407522\pi\)
\(42\) 0 0
\(43\) − 395.563i − 0.213933i −0.994263 − 0.106967i \(-0.965886\pi\)
0.994263 − 0.106967i \(-0.0341138\pi\)
\(44\) 0 0
\(45\) −72.4256 + 125.445i −0.0357657 + 0.0619481i
\(46\) 0 0
\(47\) −722.295 + 417.017i −0.326978 + 0.188781i −0.654499 − 0.756063i \(-0.727120\pi\)
0.327520 + 0.944844i \(0.393787\pi\)
\(48\) 0 0
\(49\) −1311.64 + 2011.07i −0.546290 + 0.837596i
\(50\) 0 0
\(51\) 1957.29 − 1130.04i 0.752515 − 0.434465i
\(52\) 0 0
\(53\) −1238.73 + 2145.55i −0.440988 + 0.763813i −0.997763 − 0.0668500i \(-0.978705\pi\)
0.556775 + 0.830663i \(0.312038\pi\)
\(54\) 0 0
\(55\) 172.771i 0.0571143i
\(56\) 0 0
\(57\) −609.374 −0.187557
\(58\) 0 0
\(59\) 1185.48 + 684.435i 0.340556 + 0.196620i 0.660518 − 0.750810i \(-0.270337\pi\)
−0.319962 + 0.947430i \(0.603670\pi\)
\(60\) 0 0
\(61\) 2147.71 + 3719.94i 0.577185 + 0.999714i 0.995800 + 0.0915504i \(0.0291823\pi\)
−0.418615 + 0.908164i \(0.637484\pi\)
\(62\) 0 0
\(63\) −1489.50 + 2425.27i −0.375283 + 0.611054i
\(64\) 0 0
\(65\) −84.0678 − 145.610i −0.0198977 − 0.0344638i
\(66\) 0 0
\(67\) 6589.71 + 3804.57i 1.46797 + 0.847531i 0.999356 − 0.0358732i \(-0.0114212\pi\)
0.468611 + 0.883405i \(0.344755\pi\)
\(68\) 0 0
\(69\) −2609.67 −0.548136
\(70\) 0 0
\(71\) 2869.42i 0.569216i 0.958644 + 0.284608i \(0.0918634\pi\)
−0.958644 + 0.284608i \(0.908137\pi\)
\(72\) 0 0
\(73\) −1728.76 + 2994.30i −0.324406 + 0.561889i −0.981392 − 0.192015i \(-0.938498\pi\)
0.656986 + 0.753903i \(0.271831\pi\)
\(74\) 0 0
\(75\) 2565.26 − 1481.05i 0.456045 − 0.263298i
\(76\) 0 0
\(77\) −92.2064 + 3393.48i −0.0155518 + 0.572353i
\(78\) 0 0
\(79\) −3969.87 + 2292.00i −0.636095 + 0.367250i −0.783109 − 0.621885i \(-0.786367\pi\)
0.147014 + 0.989134i \(0.453034\pi\)
\(80\) 0 0
\(81\) −758.843 + 1314.36i −0.115660 + 0.200329i
\(82\) 0 0
\(83\) − 12450.4i − 1.80729i −0.428281 − 0.903646i \(-0.640881\pi\)
0.428281 − 0.903646i \(-0.359119\pi\)
\(84\) 0 0
\(85\) −1177.40 −0.162961
\(86\) 0 0
\(87\) 2098.75 + 1211.71i 0.277283 + 0.160089i
\(88\) 0 0
\(89\) 1412.52 + 2446.55i 0.178326 + 0.308869i 0.941307 − 0.337551i \(-0.109599\pi\)
−0.762982 + 0.646420i \(0.776265\pi\)
\(90\) 0 0
\(91\) −1573.51 − 2904.86i −0.190014 − 0.350787i
\(92\) 0 0
\(93\) −1463.11 − 2534.19i −0.169166 − 0.293004i
\(94\) 0 0
\(95\) 274.924 + 158.727i 0.0304624 + 0.0175875i
\(96\) 0 0
\(97\) −6275.53 −0.666971 −0.333485 − 0.942755i \(-0.608225\pi\)
−0.333485 + 0.942755i \(0.608225\pi\)
\(98\) 0 0
\(99\) 4024.12i 0.410583i
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.r.b.95.3 yes 10
4.3 odd 2 112.5.r.a.95.3 yes 10
7.2 even 3 112.5.r.a.79.3 ✓ 10
7.3 odd 6 784.5.d.l.687.6 10
7.4 even 3 784.5.d.k.687.5 10
28.3 even 6 784.5.d.l.687.5 10
28.11 odd 6 784.5.d.k.687.6 10
28.23 odd 6 inner 112.5.r.b.79.3 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.r.a.79.3 ✓ 10 7.2 even 3
112.5.r.a.95.3 yes 10 4.3 odd 2
112.5.r.b.79.3 yes 10 28.23 odd 6 inner
112.5.r.b.95.3 yes 10 1.1 even 1 trivial
784.5.d.k.687.5 10 7.4 even 3
784.5.d.k.687.6 10 28.11 odd 6
784.5.d.l.687.5 10 28.3 even 6
784.5.d.l.687.6 10 7.3 odd 6