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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 43.18
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.18

$q$-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+(-1.44740 + 3.72895i) q^{2} +(8.25365 + 8.25365i) q^{3} +(-11.8101 - 10.7945i) q^{4} +(-26.0103 - 26.0103i) q^{5} +(-42.7237 + 18.8311i) q^{6} -18.5203 q^{7} +(57.3462 - 28.4151i) q^{8} +55.2453i q^{9} +(134.638 - 59.3438i) q^{10} +(94.1170 - 94.1170i) q^{11} +(-8.38176 - 186.570i) q^{12} +(49.3887 - 49.3887i) q^{13} +(26.8062 - 69.0610i) q^{14} -429.360i q^{15} +(22.9555 + 254.969i) q^{16} +304.665 q^{17} +(-206.007 - 79.9621i) q^{18} +(-304.164 - 304.164i) q^{19} +(26.4141 + 587.954i) q^{20} +(-152.860 - 152.860i) q^{21} +(214.732 + 487.182i) q^{22} -590.523 q^{23} +(707.843 + 238.787i) q^{24} +728.075i q^{25} +(112.683 + 255.653i) q^{26} +(212.570 - 212.570i) q^{27} +(218.726 + 199.918i) q^{28} +(391.562 - 391.562i) q^{29} +(1601.06 + 621.456i) q^{30} -1085.73i q^{31} +(-983.990 - 283.442i) q^{32} +1553.62 q^{33} +(-440.972 + 1136.08i) q^{34} +(481.718 + 481.718i) q^{35} +(596.348 - 652.451i) q^{36} +(-565.026 - 565.026i) q^{37} +(1574.46 - 693.965i) q^{38} +815.273 q^{39} +(-2230.68 - 752.507i) q^{40} +1411.17i q^{41} +(791.254 - 348.756i) q^{42} +(1117.81 - 1117.81i) q^{43} +(-2127.48 + 95.5779i) q^{44} +(1436.95 - 1436.95i) q^{45} +(854.723 - 2202.03i) q^{46} -1177.75i q^{47} +(-1914.95 + 2293.89i) q^{48} +343.000 q^{49} +(-2714.95 - 1053.82i) q^{50} +(2514.60 + 2514.60i) q^{51} +(-1116.41 + 50.1553i) q^{52} +(-1140.26 - 1140.26i) q^{53} +(484.988 + 1100.34i) q^{54} -4896.03 q^{55} +(-1062.07 + 526.255i) q^{56} -5020.93i q^{57} +(893.366 + 2026.86i) q^{58} +(-3769.63 + 3769.63i) q^{59} +(-4634.75 + 5070.77i) q^{60} +(236.168 - 236.168i) q^{61} +(4048.63 + 1571.49i) q^{62} -1023.16i q^{63} +(2481.17 - 3258.99i) q^{64} -2569.23 q^{65} +(-2248.70 + 5793.35i) q^{66} +(5703.58 + 5703.58i) q^{67} +(-3598.12 - 3288.72i) q^{68} +(-4873.97 - 4873.97i) q^{69} +(-2493.54 + 1099.06i) q^{70} -5074.17 q^{71} +(1569.80 + 3168.11i) q^{72} -9455.13i q^{73} +(2924.77 - 1289.13i) q^{74} +(-6009.27 + 6009.27i) q^{75} +(308.886 + 6875.52i) q^{76} +(-1743.07 + 1743.07i) q^{77} +(-1180.03 + 3040.11i) q^{78} -6471.21i q^{79} +(6034.74 - 7228.90i) q^{80} +7983.83 q^{81} +(-5262.18 - 2042.53i) q^{82} +(-8996.53 - 8996.53i) q^{83} +(155.232 + 3455.33i) q^{84} +(-7924.44 - 7924.44i) q^{85} +(2550.33 + 5786.15i) q^{86} +6463.63 q^{87} +(2722.91 - 8071.59i) q^{88} +3447.54i q^{89} +(3278.47 + 7438.15i) q^{90} +(-914.691 + 914.691i) q^{91} +(6974.12 + 6374.43i) q^{92} +(8961.23 - 8961.23i) q^{93} +(4391.75 + 1704.67i) q^{94} +15822.8i q^{95} +(-5782.08 - 10460.9i) q^{96} +6101.05 q^{97} +(-496.458 + 1279.03i) q^{98} +(5199.53 + 5199.53i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.44740 + 3.72895i −0.361850 + 0.932236i
\(3\) 8.25365 + 8.25365i 0.917072 + 0.917072i 0.996815 − 0.0797437i \(-0.0254102\pi\)
−0.0797437 + 0.996815i \(0.525410\pi\)
\(4\) −11.8101 − 10.7945i −0.738129 − 0.674659i
\(5\) −26.0103 − 26.0103i −1.04041 − 1.04041i −0.999148 − 0.0412651i \(-0.986861\pi\)
−0.0412651 − 0.999148i \(-0.513139\pi\)
\(6\) −42.7237 + 18.8311i −1.18677 + 0.523085i
\(7\) −18.5203 −0.377964
\(8\) 57.3462 − 28.4151i 0.896034 − 0.443986i
\(9\) 55.2453i 0.682041i
\(10\) 134.638 − 59.3438i 1.34638 − 0.593438i
\(11\) 94.1170 − 94.1170i 0.777827 − 0.777827i −0.201634 − 0.979461i \(-0.564625\pi\)
0.979461 + 0.201634i \(0.0646252\pi\)
\(12\) −8.38176 − 186.570i −0.0582066 − 1.29563i
\(13\) 49.3887 − 49.3887i 0.292241 − 0.292241i −0.545724 − 0.837965i \(-0.683745\pi\)
0.837965 + 0.545724i \(0.183745\pi\)
\(14\) 26.8062 − 69.0610i 0.136766 − 0.352352i
\(15\) − 429.360i − 1.90827i
\(16\) 22.9555 + 254.969i 0.0896698 + 0.995972i
\(17\) 304.665 1.05420 0.527102 − 0.849802i \(-0.323278\pi\)
0.527102 + 0.849802i \(0.323278\pi\)
\(18\) −206.007 − 79.9621i −0.635823 − 0.246796i
\(19\) −304.164 − 304.164i −0.842561 − 0.842561i 0.146631 − 0.989191i \(-0.453157\pi\)
−0.989191 + 0.146631i \(0.953157\pi\)
\(20\) 26.4141 + 587.954i 0.0660351 + 1.46988i
\(21\) −152.860 − 152.860i −0.346621 − 0.346621i
\(22\) 214.732 + 487.182i 0.443662 + 1.00657i
\(23\) −590.523 −1.11630 −0.558151 − 0.829740i \(-0.688489\pi\)
−0.558151 + 0.829740i \(0.688489\pi\)
\(24\) 707.843 + 238.787i 1.22889 + 0.414561i
\(25\) 728.075i 1.16492i
\(26\) 112.683 + 255.653i 0.166690 + 0.378185i
\(27\) 212.570 − 212.570i 0.291591 − 0.291591i
\(28\) 218.726 + 199.918i 0.278987 + 0.254997i
\(29\) 391.562 − 391.562i 0.465591 − 0.465591i −0.434892 − 0.900483i \(-0.643213\pi\)
0.900483 + 0.434892i \(0.143213\pi\)
\(30\) 1601.06 + 621.456i 1.77896 + 0.690506i
\(31\) − 1085.73i − 1.12979i −0.825162 − 0.564896i \(-0.808916\pi\)
0.825162 − 0.564896i \(-0.191084\pi\)
\(32\) −983.990 − 283.442i −0.960928 − 0.276799i
\(33\) 1553.62 1.42665
\(34\) −440.972 + 1136.08i −0.381464 + 0.982768i
\(35\) 481.718 + 481.718i 0.393239 + 0.393239i
\(36\) 596.348 − 652.451i 0.460145 − 0.503435i
\(37\) −565.026 − 565.026i −0.412729 − 0.412729i 0.469959 − 0.882688i \(-0.344269\pi\)
−0.882688 + 0.469959i \(0.844269\pi\)
\(38\) 1574.46 − 693.965i 1.09035 − 0.480585i
\(39\) 815.273 0.536011
\(40\) −2230.68 − 752.507i −1.39417 − 0.470317i
\(41\) 1411.17i 0.839482i 0.907644 + 0.419741i \(0.137879\pi\)
−0.907644 + 0.419741i \(0.862121\pi\)
\(42\) 791.254 − 348.756i 0.448557 − 0.197708i
\(43\) 1117.81 − 1117.81i 0.604546 − 0.604546i −0.336969 − 0.941516i \(-0.609402\pi\)
0.941516 + 0.336969i \(0.109402\pi\)
\(44\) −2127.48 + 95.5779i −1.09890 + 0.0493687i
\(45\) 1436.95 − 1436.95i 0.709605 − 0.709605i
\(46\) 854.723 − 2202.03i 0.403934 − 1.04066i
\(47\) − 1177.75i − 0.533158i −0.963813 − 0.266579i \(-0.914107\pi\)
0.963813 − 0.266579i \(-0.0858933\pi\)
\(48\) −1914.95 + 2293.89i −0.831144 + 0.995611i
\(49\) 343.000 0.142857
\(50\) −2714.95 − 1053.82i −1.08598 − 0.421526i
\(51\) 2514.60 + 2514.60i 0.966782 + 0.966782i
\(52\) −1116.41 + 50.1553i −0.412874 + 0.0185486i
\(53\) −1140.26 − 1140.26i −0.405932 − 0.405932i 0.474385 − 0.880317i \(-0.342670\pi\)
−0.880317 + 0.474385i \(0.842670\pi\)
\(54\) 484.988 + 1100.34i 0.166320 + 0.377344i
\(55\) −4896.03 −1.61852
\(56\) −1062.07 + 526.255i −0.338669 + 0.167811i
\(57\) − 5020.93i − 1.54538i
\(58\) 893.366 + 2026.86i 0.265567 + 0.602515i
\(59\) −3769.63 + 3769.63i −1.08291 + 1.08291i −0.0866785 + 0.996236i \(0.527625\pi\)
−0.996236 + 0.0866785i \(0.972375\pi\)
\(60\) −4634.75 + 5070.77i −1.28743 + 1.40855i
\(61\) 236.168 − 236.168i 0.0634691 − 0.0634691i −0.674660 − 0.738129i \(-0.735710\pi\)
0.738129 + 0.674660i \(0.235710\pi\)
\(62\) 4048.63 + 1571.49i 1.05323 + 0.408815i
\(63\) − 1023.16i − 0.257787i
\(64\) 2481.17 − 3258.99i 0.605753 − 0.795652i
\(65\) −2569.23 −0.608102
\(66\) −2248.70 + 5793.35i −0.516232 + 1.32997i
\(67\) 5703.58 + 5703.58i 1.27057 + 1.27057i 0.945790 + 0.324779i \(0.105290\pi\)
0.324779 + 0.945790i \(0.394710\pi\)
\(68\) −3598.12 − 3288.72i −0.778140 − 0.711229i
\(69\) −4873.97 − 4873.97i −1.02373 − 1.02373i
\(70\) −2493.54 + 1099.06i −0.508886 + 0.224298i
\(71\) −5074.17 −1.00658 −0.503290 − 0.864117i \(-0.667877\pi\)
−0.503290 + 0.864117i \(0.667877\pi\)
\(72\) 1569.80 + 3168.11i 0.302816 + 0.611132i
\(73\) − 9455.13i − 1.77428i −0.461501 − 0.887140i \(-0.652689\pi\)
0.461501 − 0.887140i \(-0.347311\pi\)
\(74\) 2924.77 − 1289.13i 0.534107 − 0.235415i
\(75\) −6009.27 + 6009.27i −1.06831 + 1.06831i
\(76\) 308.886 + 6875.52i 0.0534774 + 1.19036i
\(77\) −1743.07 + 1743.07i −0.293991 + 0.293991i
\(78\) −1180.03 + 3040.11i −0.193956 + 0.499689i
\(79\) − 6471.21i − 1.03689i −0.855112 − 0.518443i \(-0.826512\pi\)
0.855112 − 0.518443i \(-0.173488\pi\)
\(80\) 6034.74 − 7228.90i 0.942928 − 1.12952i
\(81\) 7983.83 1.21686
\(82\) −5262.18 − 2042.53i −0.782596 − 0.303767i
\(83\) −8996.53 − 8996.53i −1.30593 − 1.30593i −0.924328 − 0.381599i \(-0.875374\pi\)
−0.381599 − 0.924328i \(-0.624626\pi\)
\(84\) 155.232 + 3455.33i 0.0220000 + 0.489702i
\(85\) −7924.44 − 7924.44i −1.09681 − 1.09681i
\(86\) 2550.33 + 5786.15i 0.344825 + 0.782335i
\(87\) 6463.63 0.853960
\(88\) 2722.91 − 8071.59i 0.351615 − 1.04230i
\(89\) 3447.54i 0.435241i 0.976033 + 0.217621i \(0.0698295\pi\)
−0.976033 + 0.217621i \(0.930170\pi\)
\(90\) 3278.47 + 7438.15i 0.404749 + 0.918290i
\(91\) −914.691 + 914.691i −0.110457 + 0.110457i
\(92\) 6974.12 + 6374.43i 0.823975 + 0.753123i
\(93\) 8961.23 − 8961.23i 1.03610 − 1.03610i
\(94\) 4391.75 + 1704.67i 0.497029 + 0.192923i
\(95\) 15822.8i 1.75322i
\(96\) −5782.08 − 10460.9i −0.627396 − 1.13508i
\(97\) 6101.05 0.648427 0.324214 − 0.945984i \(-0.394900\pi\)
0.324214 + 0.945984i \(0.394900\pi\)
\(98\) −496.458 + 1279.03i −0.0516928 + 0.133177i
\(99\) 5199.53 + 5199.53i 0.530510 + 0.530510i
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.43.18 ✓ 96
4.3 odd 2 448.5.k.a.15.9 96
16.3 odd 4 inner 112.5.k.a.99.18 yes 96
16.13 even 4 448.5.k.a.239.9 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.18 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.18 yes 96 16.3 odd 4 inner
448.5.k.a.15.9 96 4.3 odd 2
448.5.k.a.239.9 96 16.13 even 4