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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 79.3
Root \(0.500000 - 3.92097i\) of defining polynomial
Character \(\chi\) \(=\) 112.79
Dual form 112.5.r.b.95.3

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

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

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 −0.251680 − 0.967811i \(-0.580983\pi\)
0.712309 + 0.701866i \(0.247650\pi\)
\(4\) 0 0
\(5\) −1.24690 + 2.15969i −0.0498759 + 0.0863876i −0.889886 − 0.456184i \(-0.849216\pi\)
0.840010 + 0.542571i \(0.182549\pi\)
\(6\) 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 −0.727000 − 0.686637i \(-0.759086\pi\)
0.231145 + 0.972919i \(0.425753\pi\)
\(12\) 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 0.908003 + 0.418963i \(0.137606\pi\)
−0.0911692 + 0.995835i \(0.529060\pi\)
\(18\) 0 0
\(19\) −110.243 − 63.6489i −0.305383 − 0.176313i 0.339476 − 0.940615i \(-0.389750\pi\)
−0.644858 + 0.764302i \(0.723084\pi\)
\(20\) 0 0
\(21\) 6.37111 + 234.476i 0.0144470 + 0.531692i
\(22\) 0 0
\(23\) −472.122 − 272.580i −0.892479 − 0.515273i −0.0177267 − 0.999843i \(-0.505643\pi\)
−0.874753 + 0.484570i \(0.838976\pi\)
\(24\) 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 −0.749475 − 0.662033i \(-0.769694\pi\)
0.198600 + 0.980081i \(0.436360\pi\)
\(32\) 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 0.497011 + 0.867744i \(0.334431\pi\)
−0.999994 + 0.00344766i \(0.998903\pi\)
\(38\) 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 \(0.0341138\pi\)
−0.994263 + 0.106967i \(0.965886\pi\)
\(44\) 0 0
\(45\) −72.4256 − 125.445i −0.0357657 − 0.0619481i
\(46\) 0 0
\(47\) −722.295 − 417.017i −0.326978 − 0.188781i 0.327520 − 0.944844i \(-0.393787\pi\)
−0.654499 + 0.756063i \(0.727120\pi\)
\(48\) 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 0.556775 − 0.830663i \(-0.312038\pi\)
−0.997763 + 0.0668500i \(0.978705\pi\)
\(54\) 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 −0.319962 − 0.947430i \(-0.603670\pi\)
0.660518 + 0.750810i \(0.270337\pi\)
\(60\) 0 0
\(61\) 2147.71 − 3719.94i 0.577185 − 0.999714i −0.418615 − 0.908164i \(-0.637484\pi\)
0.995800 − 0.0915504i \(-0.0291823\pi\)
\(62\) 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 0.468611 − 0.883405i \(-0.344755\pi\)
0.999356 + 0.0358732i \(0.0114212\pi\)
\(68\) 0 0
\(69\) −2609.67 −0.548136
\(70\) 0 0
\(71\) − 2869.42i − 0.569216i −0.958644 − 0.284608i \(-0.908137\pi\)
0.958644 − 0.284608i \(-0.0918634\pi\)
\(72\) 0 0
\(73\) −1728.76 − 2994.30i −0.324406 − 0.561889i 0.656986 − 0.753903i \(-0.271831\pi\)
−0.981392 + 0.192015i \(0.938498\pi\)
\(74\) 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 0.147014 − 0.989134i \(-0.453034\pi\)
−0.783109 + 0.621885i \(0.786367\pi\)
\(80\) 0 0
\(81\) −758.843 − 1314.36i −0.115660 − 0.200329i
\(82\) 0 0
\(83\) 12450.4i 1.80729i 0.428281 + 0.903646i \(0.359119\pi\)
−0.428281 + 0.903646i \(0.640881\pi\)
\(84\) 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 −0.762982 − 0.646420i \(-0.776265\pi\)
0.941307 + 0.337551i \(0.109599\pi\)
\(90\) 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.79.3 yes 10
4.3 odd 2 112.5.r.a.79.3 ✓ 10
7.2 even 3 784.5.d.k.687.6 10
7.4 even 3 112.5.r.a.95.3 yes 10
7.5 odd 6 784.5.d.l.687.5 10
28.11 odd 6 inner 112.5.r.b.95.3 yes 10
28.19 even 6 784.5.d.l.687.6 10
28.23 odd 6 784.5.d.k.687.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.r.a.79.3 ✓ 10 4.3 odd 2
112.5.r.a.95.3 yes 10 7.4 even 3
112.5.r.b.79.3 yes 10 1.1 even 1 trivial
112.5.r.b.95.3 yes 10 28.11 odd 6 inner
784.5.d.k.687.5 10 28.23 odd 6
784.5.d.k.687.6 10 7.2 even 3
784.5.d.l.687.5 10 7.5 odd 6
784.5.d.l.687.6 10 28.19 even 6