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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(43,112)] 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("112.43"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.5
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.5

$q$-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+(-3.78615 + 1.29039i) q^{2} +(11.4158 + 11.4158i) q^{3} +(12.6698 - 9.77120i) q^{4} +(8.99601 + 8.99601i) q^{5} +(-57.9529 - 28.4912i) q^{6} -18.5203 q^{7} +(-35.3610 + 53.3441i) q^{8} +179.643i q^{9} +(-45.6686 - 22.4519i) q^{10} +(16.8258 - 16.8258i) q^{11} +(256.183 + 33.0899i) q^{12} +(-139.940 + 139.940i) q^{13} +(70.1204 - 23.8983i) q^{14} +205.394i q^{15} +(65.0474 - 247.598i) q^{16} +327.145 q^{17} +(-231.809 - 680.154i) q^{18} +(134.031 + 134.031i) q^{19} +(201.879 + 26.0758i) q^{20} +(-211.424 - 211.424i) q^{21} +(-41.9931 + 85.4167i) q^{22} -352.136 q^{23} +(-1012.64 + 205.292i) q^{24} -463.143i q^{25} +(349.257 - 710.412i) q^{26} +(-1126.09 + 1126.09i) q^{27} +(-234.648 + 180.965i) q^{28} +(-533.411 + 533.411i) q^{29} +(-265.038 - 777.652i) q^{30} +1492.89i q^{31} +(73.2191 + 1021.38i) q^{32} +384.161 q^{33} +(-1238.62 + 422.144i) q^{34} +(-166.609 - 166.609i) q^{35} +(1755.33 + 2276.04i) q^{36} +(431.104 + 431.104i) q^{37} +(-680.415 - 334.510i) q^{38} -3195.08 q^{39} +(-797.993 + 161.776i) q^{40} -2824.85i q^{41} +(1073.30 + 527.664i) q^{42} +(-341.119 + 341.119i) q^{43} +(48.7711 - 377.587i) q^{44} +(-1616.07 + 1616.07i) q^{45} +(1333.24 - 454.392i) q^{46} -3999.13i q^{47} +(3569.11 - 2083.97i) q^{48} +343.000 q^{49} +(597.635 + 1753.53i) q^{50} +(3734.64 + 3734.64i) q^{51} +(-405.631 + 3140.40i) q^{52} +(524.825 + 524.825i) q^{53} +(2810.45 - 5716.65i) q^{54} +302.730 q^{55} +(654.896 - 987.947i) q^{56} +3060.16i q^{57} +(1331.27 - 2707.88i) q^{58} +(25.1602 - 25.1602i) q^{59} +(2006.95 + 2602.30i) q^{60} +(202.929 - 202.929i) q^{61} +(-1926.41 - 5652.31i) q^{62} -3327.03i q^{63} +(-1595.19 - 3772.61i) q^{64} -2517.81 q^{65} +(-1454.49 + 495.717i) q^{66} +(5554.26 + 5554.26i) q^{67} +(4144.86 - 3196.60i) q^{68} +(-4019.93 - 4019.93i) q^{69} +(845.794 + 415.814i) q^{70} +4830.28 q^{71} +(-9582.90 - 6352.36i) q^{72} -4640.36i q^{73} +(-2188.51 - 1075.93i) q^{74} +(5287.17 - 5287.17i) q^{75} +(3007.80 + 388.503i) q^{76} +(-311.618 + 311.618i) q^{77} +(12097.0 - 4122.89i) q^{78} +2215.33i q^{79} +(2812.56 - 1642.23i) q^{80} -11159.5 q^{81} +(3645.15 + 10695.3i) q^{82} +(8545.70 + 8545.70i) q^{83} +(-4744.57 - 612.834i) q^{84} +(2943.00 + 2943.00i) q^{85} +(851.350 - 1731.70i) q^{86} -12178.7 q^{87} +(302.580 + 1492.53i) q^{88} +8352.39i q^{89} +(4033.32 - 8204.04i) q^{90} +(2591.73 - 2591.73i) q^{91} +(-4461.49 + 3440.79i) q^{92} +(-17042.6 + 17042.6i) q^{93} +(5160.43 + 15141.3i) q^{94} +2411.50i q^{95} +(-10824.0 + 12495.8i) q^{96} +12536.7 q^{97} +(-1298.65 + 442.603i) q^{98} +(3022.63 + 3022.63i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 6 q^{4} - 132 q^{6} - 200 q^{10} - 96 q^{11} + 660 q^{12} - 294 q^{14} + 642 q^{16} - 810 q^{18} + 1408 q^{19} - 2604 q^{20} - 106 q^{22} - 1152 q^{23} + 3880 q^{24} + 6828 q^{26} + 3648 q^{27} - 864 q^{29}+ \cdots - 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.78615 + 1.29039i −0.946536 + 0.322597i
\(3\) 11.4158 + 11.4158i 1.26843 + 1.26843i 0.946901 + 0.321527i \(0.104196\pi\)
0.321527 + 0.946901i \(0.395804\pi\)
\(4\) 12.6698 − 9.77120i 0.791862 − 0.610700i
\(5\) 8.99601 + 8.99601i 0.359841 + 0.359841i 0.863754 − 0.503914i \(-0.168107\pi\)
−0.503914 + 0.863754i \(0.668107\pi\)
\(6\) −57.9529 − 28.4912i −1.60980 − 0.791421i
\(7\) −18.5203 −0.377964
\(8\) −35.3610 + 53.3441i −0.552516 + 0.833502i
\(9\) 179.643i 2.21781i
\(10\) −45.6686 − 22.4519i −0.456686 − 0.224519i
\(11\) 16.8258 − 16.8258i 0.139056 − 0.139056i −0.634152 − 0.773208i \(-0.718651\pi\)
0.773208 + 0.634152i \(0.218651\pi\)
\(12\) 256.183 + 33.0899i 1.77905 + 0.229791i
\(13\) −139.940 + 139.940i −0.828050 + 0.828050i −0.987247 − 0.159197i \(-0.949109\pi\)
0.159197 + 0.987247i \(0.449109\pi\)
\(14\) 70.1204 − 23.8983i 0.357757 − 0.121930i
\(15\) 205.394i 0.912863i
\(16\) 65.0474 − 247.598i 0.254091 − 0.967180i
\(17\) 327.145 1.13199 0.565995 − 0.824409i \(-0.308492\pi\)
0.565995 + 0.824409i \(0.308492\pi\)
\(18\) −231.809 − 680.154i −0.715461 − 2.09924i
\(19\) 134.031 + 134.031i 0.371278 + 0.371278i 0.867943 − 0.496665i \(-0.165442\pi\)
−0.496665 + 0.867943i \(0.665442\pi\)
\(20\) 201.879 + 26.0758i 0.504699 + 0.0651895i
\(21\) −211.424 − 211.424i −0.479420 − 0.479420i
\(22\) −41.9931 + 85.4167i −0.0867626 + 0.176481i
\(23\) −352.136 −0.665663 −0.332832 − 0.942986i \(-0.608004\pi\)
−0.332832 + 0.942986i \(0.608004\pi\)
\(24\) −1012.64 + 205.292i −1.75806 + 0.356410i
\(25\) − 463.143i − 0.741030i
\(26\) 349.257 − 710.412i 0.516653 − 1.05091i
\(27\) −1126.09 + 1126.09i −1.54471 + 1.54471i
\(28\) −234.648 + 180.965i −0.299296 + 0.230823i
\(29\) −533.411 + 533.411i −0.634259 + 0.634259i −0.949133 − 0.314875i \(-0.898038\pi\)
0.314875 + 0.949133i \(0.398038\pi\)
\(30\) −265.038 − 777.652i −0.294487 − 0.864058i
\(31\) 1492.89i 1.55348i 0.629823 + 0.776739i \(0.283128\pi\)
−0.629823 + 0.776739i \(0.716872\pi\)
\(32\) 73.2191 + 1021.38i 0.0715031 + 0.997440i
\(33\) 384.161 0.352765
\(34\) −1238.62 + 422.144i −1.07147 + 0.365177i
\(35\) −166.609 − 166.609i −0.136007 − 0.136007i
\(36\) 1755.33 + 2276.04i 1.35442 + 1.75620i
\(37\) 431.104 + 431.104i 0.314904 + 0.314904i 0.846806 − 0.531902i \(-0.178522\pi\)
−0.531902 + 0.846806i \(0.678522\pi\)
\(38\) −680.415 − 334.510i −0.471201 − 0.231655i
\(39\) −3195.08 −2.10064
\(40\) −797.993 + 161.776i −0.498746 + 0.101110i
\(41\) − 2824.85i − 1.68046i −0.542233 − 0.840228i \(-0.682421\pi\)
0.542233 − 0.840228i \(-0.317579\pi\)
\(42\) 1073.30 + 527.664i 0.608448 + 0.299129i
\(43\) −341.119 + 341.119i −0.184488 + 0.184488i −0.793308 − 0.608820i \(-0.791643\pi\)
0.608820 + 0.793308i \(0.291643\pi\)
\(44\) 48.7711 − 377.587i 0.0251917 − 0.195035i
\(45\) −1616.07 + 1616.07i −0.798060 + 0.798060i
\(46\) 1333.24 − 454.392i 0.630075 − 0.214741i
\(47\) − 3999.13i − 1.81038i −0.425009 − 0.905189i \(-0.639729\pi\)
0.425009 − 0.905189i \(-0.360271\pi\)
\(48\) 3569.11 − 2083.97i 1.54909 − 0.904501i
\(49\) 343.000 0.142857
\(50\) 597.635 + 1753.53i 0.239054 + 0.701411i
\(51\) 3734.64 + 3734.64i 1.43585 + 1.43585i
\(52\) −405.631 + 3140.40i −0.150011 + 1.16139i
\(53\) 524.825 + 524.825i 0.186837 + 0.186837i 0.794327 − 0.607490i \(-0.207824\pi\)
−0.607490 + 0.794327i \(0.707824\pi\)
\(54\) 2810.45 − 5716.65i 0.963804 − 1.96044i
\(55\) 302.730 0.100076
\(56\) 654.896 − 987.947i 0.208831 − 0.315034i
\(57\) 3060.16i 0.941878i
\(58\) 1331.27 − 2707.88i 0.395739 − 0.804959i
\(59\) 25.1602 − 25.1602i 0.00722788 − 0.00722788i −0.703484 − 0.710711i \(-0.748373\pi\)
0.710711 + 0.703484i \(0.248373\pi\)
\(60\) 2006.95 + 2602.30i 0.557485 + 0.722862i
\(61\) 202.929 − 202.929i 0.0545361 − 0.0545361i −0.679313 − 0.733849i \(-0.737722\pi\)
0.733849 + 0.679313i \(0.237722\pi\)
\(62\) −1926.41 − 5652.31i −0.501148 − 1.47042i
\(63\) − 3327.03i − 0.838255i
\(64\) −1595.19 − 3772.61i −0.389452 − 0.921047i
\(65\) −2517.81 −0.595932
\(66\) −1454.49 + 495.717i −0.333905 + 0.113801i
\(67\) 5554.26 + 5554.26i 1.23731 + 1.23731i 0.961098 + 0.276207i \(0.0890776\pi\)
0.276207 + 0.961098i \(0.410922\pi\)
\(68\) 4144.86 − 3196.60i 0.896380 − 0.691306i
\(69\) −4019.93 − 4019.93i −0.844345 − 0.844345i
\(70\) 845.794 + 415.814i 0.172611 + 0.0848601i
\(71\) 4830.28 0.958198 0.479099 − 0.877761i \(-0.340963\pi\)
0.479099 + 0.877761i \(0.340963\pi\)
\(72\) −9582.90 − 6352.36i −1.84855 − 1.22538i
\(73\) − 4640.36i − 0.870775i −0.900243 − 0.435387i \(-0.856611\pi\)
0.900243 − 0.435387i \(-0.143389\pi\)
\(74\) −2188.51 − 1075.93i −0.399655 − 0.196481i
\(75\) 5287.17 − 5287.17i 0.939942 − 0.939942i
\(76\) 3007.80 + 388.503i 0.520740 + 0.0672615i
\(77\) −311.618 + 311.618i −0.0525583 + 0.0525583i
\(78\) 12097.0 − 4122.89i 1.98833 − 0.677661i
\(79\) 2215.33i 0.354963i 0.984124 + 0.177482i \(0.0567950\pi\)
−0.984124 + 0.177482i \(0.943205\pi\)
\(80\) 2812.56 − 1642.23i 0.439463 − 0.256598i
\(81\) −11159.5 −1.70089
\(82\) 3645.15 + 10695.3i 0.542110 + 1.59061i
\(83\) 8545.70 + 8545.70i 1.24048 + 1.24048i 0.959800 + 0.280684i \(0.0905613\pi\)
0.280684 + 0.959800i \(0.409439\pi\)
\(84\) −4744.57 − 612.834i −0.672417 − 0.0868528i
\(85\) 2943.00 + 2943.00i 0.407336 + 0.407336i
\(86\) 851.350 − 1731.70i 0.115110 − 0.234140i
\(87\) −12178.7 −1.60902
\(88\) 302.580 + 1492.53i 0.0390728 + 0.192734i
\(89\) 8352.39i 1.05446i 0.849722 + 0.527231i \(0.176770\pi\)
−0.849722 + 0.527231i \(0.823230\pi\)
\(90\) 4033.32 − 8204.04i 0.497941 − 1.01284i
\(91\) 2591.73 − 2591.73i 0.312973 − 0.312973i
\(92\) −4461.49 + 3440.79i −0.527114 + 0.406521i
\(93\) −17042.6 + 17042.6i −1.97047 + 1.97047i
\(94\) 5160.43 + 15141.3i 0.584023 + 1.71359i
\(95\) 2411.50i 0.267202i
\(96\) −10824.0 + 12495.8i −1.17448 + 1.35588i
\(97\) 12536.7 1.33241 0.666207 − 0.745767i \(-0.267917\pi\)
0.666207 + 0.745767i \(0.267917\pi\)
\(98\) −1298.65 + 442.603i −0.135219 + 0.0460853i
\(99\) 3022.63 + 3022.63i 0.308401 + 0.308401i
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.k.a.43.5 ✓ 96
4.3 odd 2 448.5.k.a.15.2 96
16.3 odd 4 inner 112.5.k.a.99.5 yes 96
16.13 even 4 448.5.k.a.239.2 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.5 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.5 yes 96 16.3 odd 4 inner
448.5.k.a.15.2 96 4.3 odd 2
448.5.k.a.239.2 96 16.13 even 4